NCO:ncap2 is the function to do it:
ncap2 -s 'new_var=var1+var2' in_filename.nc out_filename.nc
The output file will have all of the variables that exist in the input file as well as the new_var. Add -O if your input and output files are the same (overwrite).
I do not know what the -s stands for.
BUT the new_var will have the same long_name as the first variable used for summing (i.e. it could make some things a bit confusing). To change it, use a very complicated (but allegedly also very powerful) NCO:ncatted. Fortunately, its documentation has just the right example:
Change the value of the
long_name attribute for variable
T from whatever it currently is to “temperature”:
ncatted -a long_name,T,o,c,temperature in.nc