You need to replace a part called a "Tilt and Turn Dirve Gear". This is the gearing part running down the edge of the window, the handle then connects into it. The "drive gear" has a cog in the middle as you describe.
Once this part is replaced - hopefully nothing else is broken and the window should work again. The following company offer a good selection:
https://www.upvc-hardware.co.uk/locks-gearing-163/drive-gear-mechanisms-187/Fitting it is a bit complicated for someone without experiance. I'll try to explain here, if you don't understand then best call out a double glazing repair company.
Remove the handle (2 screws on the front of the handle usually hidden behing a cover plate)
Remove the screws from down the side of the window that fasten the drive gear to the window
Disconnect the drive gear top and bottom from the rest of the gearing.
Cut the new drive gear to the same length as the original.
Put in place and connect with gearing top and bottom.
Put screws in down the side to hold drive gear in place and re-fit handle
When connecting the new drive gear top and bottom, make sure the roller cam locking points all around the window are in the same position as the cams on the new drive gear so that its all in sync.
Hope you get it sorted Scott
