I’ve seen this done at several shops. The most common approach is to use double-colon rules, assuming you’re using something like GNU make. In your common makefile you would have something like this:
clean::
# standard cleanup, like remove all .o's:
rm -f *.o
Note that there are two colons following clean, not just one!
In your other makefile you just declare clean again, as a double-colon rule:
clean::
# custom cleanup, like remove my special generated files:
rm -f *.h.gen
When you invoke make clean, GNU make will automagically run both of these “branches” of the clean rule:
% make clean
rm -f *.o
rm -f *.h.gen
It’s simple to set up and it composes quite neatly I think. Note that specifically because it is a double-colon rule, you don’t get the “overriding commands” errors you normally get when you define two rules for the same target. That’s sort of the point of double-colon rules.