IMHO, the French could have easily won Agincourt for two reasons:
1. Henry's men, were already very weary from hunger, illness and marching. Even though he knew as well as the French did that his army would perform better on the defensive, Henry was eventually forced to take a calculated risk, and move his army further forward to start the battle. This entailed abandoning his chosen position and pulling out, advancing, and then re-installing the long sharpened wooden stakes pointed outwards toward the enemy which helped protect the longbowmen from cavalry charges. If the French cavalry had charged before the stakes had been hammered back in, the result would have been disastrous for the English, as it was at the Battle of Patay. However, the French seem to have been caught off guard by the English advance.
2. What the French endured going into the battle was a narrow path completely soaked from heavy rains the night before. The path was too narrow to fit that many men so they were gathered very closely. The narrow path and thick mud were the prime reason Henry chose this as his new battle ground. The 1,000–1,500 French men-at-arms are described as shoulder to shoulder and four deep, which implies a tight line about 250–300 men long. The Battlefield Defectives actually did a very good piece on how that many men gathered that tightly would act if say one them fell down. It basically becomes a slow moving disaster with men constantly falling over one another while being showered with volleys of English arrows.
While this is an awful scenario to be in Charles d'Albret could afford the men. He had a very important advantage in this fight. That advantage shattered when his men were not able to reform at the end of the path. What really happens to a unit under heavy fire is that random men within the unit get taken out. This means that the unit loses its shape and vulnerable form gaps where enemy infantry can strike. Had the men been able to cross the death march and reform, their numerical advantage would have been more than enough to overcome Henry's logistically suffering men.
However they did not. The few men who did cross the pass were not able to reform their ranks to build a fighting front. They instead routed. They turned and ran back through the killing field. Now you have a mass of men running forward and backward through a very narrow, ever more stirred up mud pit. This is the absolute best target for longbowmen who don't fire at an individual, but in mass at a group. The fact that Charles' men were not able to continue on and gather themselves, not that that would have been an easy,cost not only their lives, but the lives of thousands of other French soldiers as well.
It is also noteworthy that the French army included a detachment of Genoese mercenary crossbowmen who would have outranged the English longbowmen, but the French aristocracy in command didn't want to lead the battle with foreign soldiers, so they were never deployed. So in a sense, winning would have required the French command to deploy their troops as the English did, by their order of effectiveness, as opposed to by their social rank (knights in front, then feudal levies, then mercenaries). The French army was hamstrung by internal feudal politicking and couldn't maneuver or deploy as effectively as a result.
Putting the battle totally aside, I believe the best choice for the French would have been just to block the English - they were running out of supplies, exhausted, and wouldn't have lasted two more weeks of marching. Put a blocking force in their way so they can't reach any ports, hunt down foraging parties with cavalry, and they'd be forced to surrender.
Of course, this would lack the glamour of a glorious victory - good luck getting the French knights to agree on this course of non-action.
Fix the problems and we have a French Victory with the result that….
The leadership of the French Armagnac faction is probably mostly intact, with a more-or-less intact army and maybe some high-status English prisoners. They could try and take Harfleur back, or maybe march on Calais. Except that…
Duke John the Fearless of Burgundy is in Dijon with an army of his own. In theory, Armagnacs and Burgundians have put their differences aside to jointly repel Henry V's invasion. In reality, John was waiting and seeing what happened before he made his move. He could take advantage of the mostly-Armagnac defeat at Agincourt to march on Paris. He could sit tight or opportunistically go after Calais himself.
Thomas duke of Clarence, the slain Henry's younger brother and heir presumptive, is in Calais. He is now King Thomas I of England. He went there from Harfleur rather than join the march that led to Agincourt, allegedly sick. In reality, it is suspected that he was got out of the way by Henry because of his opposition to the march and his close personal ties to leading Armagnacs like Charles d'Orleans and Charles d'Albret. Does the new king now try to salvage what is left of England's fortunes in France by arranging a truce with his Armagnac friends, maybe to help offset any Burgundian move against English possessions.
Henry V was operating a pro-Burgundian diplomacy before he died. His next oldest brother, John duke of Bedford, was left in England as lieutenant while Henry was campaigning in France, governing with the aid of a council headed by the Archbishop of Canterbury. What does he do, especially if he trusts Thomas about as much as Henry did? Although, Thomas is now the king...
King Thomas would have his policy of caution vindicated and Henry revealed as a dead fool, and a martyr to no one. King Thomas would have to retrench, holding up in Calais, and hopefully watch the Burgundians and Armagnac factions tear each other to pieces. Thomas would never have expected to be king anyway, so just getting the crown should be enough for him.
To add further "what ifs" Suppose Henry is alive and in French captivity, so that Thomas and John can squabble over the best course of action in France and the dauphin's faction can demand a predictably massive ransom from England...
Killing a king in battle is one thing, killing him afterword is another. However, considering the Dauphin's (he was there) feelings towards Henry, Henry's survival in a lost Agincourt is highly unlikely. Also, there were a lot of doubters about Henry's invasion considering how long it had been since Charles the Wise had so effectively outmaneuvered the English from so much of France.
The vast majority of the English archers at Agincourt would have been killed either in or after the battle, that sort of trained manpower is hard to replace quickly. Now, Henry had sent a lot of sick men back to England or Calais from Harfleur, so the massacre of the army at Agincourt would not mean the destruction of the entire force he had mustered for the French campaign (maybe only half of it). Still, it would be a blow to England's immediate term military outlook.
The hundred years would have continued regardless, but the age of knights would have continued for a few decades longer, as the longbow would have to prove itself at a later date. (The loss would have been blamed on the difference in numbers of men on the field not on the weapons used)
Thomas of Lancaster, Duke of Clarence would become king of England and continue the war as soon as his army is up to scratch.
This would be necessary to clear the name of his family from the loss his brother suffered - his other brothers would demand that. His brother John of Lancaster,Duke of Bedford would not accept him as king if Thomas makes peace so soon after the loss at Agincourt.
But with the French winning, absolutism would not have been "invented", at least not in France. With so many aristocrats surviving, the kings of France would have a lot of opposition to impose their personal rule.
This would also put a serious blow to enlightenment as "commoners" like Descartes would not get a voice or not as big a voice as he had.
There is still another card to be played in the Hundred Years War. Joan of Arc. She's only 3 years old at Agincourt. But when she gets old enough, and gets herself moving, look out! It's interesting to speculate on how King Thomas would have taken Joan, though. Bedford treated her with contempt until it was too late, and then blamed it all on "witchcraft".