How to fix damage to my fence door's latch?
I would,


unscrew the latch and remove it,
apply wood glue under the broken piece,
clamp it and let it dry for an hour or two
drill some deep pilot holes through 95% of the thickness of the whole gate
reattach the latch with longer screws .


I would consider using bolts instead, with washers to spread the load on the far side.