The answer to your question depends on the rule that is used for compiling the C++ files. Since you do not want to share your makefile I explain this with the default implicit rule for C++ programs.
You can override any make variable, say 'MY_VAR' by calling make like this:
make MY_VAR='Hello'
The default recipe for compiling C++ looks similar to this:
%.o : %.cpp
$(CXX) $(CPPFLAGS) $(CXXFLAGS) -c $< -o $@
CPPFLAGS are by convention the flags passed to the C and C++ preprocessor, CXXFLAGS are relevant for C++ compilation.
So I presume you makefile sets CXXFLAGS, lets say like this:
CXXFLAGS = -O3
You want it to be -O0, so you can override CXXFLAGS.
Call make like this:
make -j20 TGT=stuff CXXFLAGS='-O0'
Since you want to debug you probably should use:
make -j20 TGT=stuff CXXFLAGS='-O0 -g'
If the default for CXXFLAGS from your makefile includes other important flags you would have to specify those too - you override its value.
See also:
https://www.gnu.org/software/make/manual/html_node/Overriding.html
The make manual is really good, I can only recommend having a look.