Gawyn was a troubled soul, and his character had some seriously dark issues. About the time Wotmania was shutting down, I had written a theory that he and the Younglings that helped Elaida in the WT split, were actually being influenced by Mesaana. (I never submitted it and no longer have it saved.) If you recall, her modus operandi in the AoL was to turn children against teachers, parents, or adult authority figures in general, thus the moniker of "Mesaana's Children." I felt the "Younglings" fit this bill in both description and their actions during that battle.

If that theory was true, and he had been influenced by Mesaana, then those circumstances could have led to Gawyn killing Egwene. As things happened after the WT split, Elaida distanced herself from Gawyn and his boys, and they became almost rogues, seldom in the WT. So, Mesaana could have lost influence with them, or they had served her purpose by insuring the split and she no longer needed them.