Check the sides of the dovetail for cracking or other damage. Weld the cracks and smooth out any burrs.
Push the round detent down and slide the broken plastic out, forward towards muzzle.
Your replacement sight will be slightly large, need to run the left and right sides against a fine file to slowly remove material evenly.
Test the fit by trying to ease the replacement sight into the dovetail slot. It should slide in with a tiny amount of drag...dont force it in as this will cause it the crack and easily break.
Remmember to hold the retaining detent down as you slide the sight into place. It should pop up and prevent the sight from sliding out when released.
The Ithaca 37/87 is a good shotgun, but it does require parts fitting to replace most stuff.