Chlorine atoms tend to form ions by taking in one electron. This means the atom has a charge of -1. How does the additional electron affect the atomic number and the
mass number for this atom?

The additional electron does not affect the atomic number nor the mass number. The atomic mass is the sum of neutrons and protons. Electrons are not accounted in atomic mass as their mass is so small, approximately 1/1,836 of a protons mass. Atomic number is the total number of protons of an atom. Chlorine taking in another electron will not change its atomic number. In an ion, the atomic number and atomic mass do not change from the original.