How do I repair decking where my screws have broken?
Using a block of wood and a large-ish hammer, drive the decking down tight at that location.
Pilot a 3/32&quot; or 1/8&quot; hole in the divot from the old screw head, but at a slight angle parallel with the joist, through the decking only.
Run in a new screw. If you find that they're still breaking, use a countersink bit to slightly deepen the divot first.

The fact is that you can probably just run a new screw in at a slight bias and it'll find its way around the old screw, but this is probably easier and more reliable for less experienced carpenters.
Be aware that it may be easy to run the screw in too deep due to the aged wood and existing hole. Go gently.