In your example, you are using typesetting. This typesetting is obfucasting the issue. Can you type out the actual code? This will make it much more clear what is happening. Write it out using the D and Derivative expressions.
Derivative expressions don't have addition defined on them. You could define that. Or you could just write out your new operator as a function:
myOperator[f_] := (Derivative[0, 1][f] + Derivative[1, 0][f]);
myOperator[f[x, y]]
