"El papá le pega mucho al niño." How would you translate that? Would it depend entirely on the context? Let's say the speaker is from Mexico.

Pegar can also have the meaning/sense of sticking (like glue), so that phrase could mean ?? something like the father is around his child a lot.
The context would help considerably, yes.

At least in spoken speach in Mexico, a spanking is usually referred to as a "nalgada" ("Le dio una nalgada," from "nalgas" or backside/bottom).
But if you are speaking about something habitual, it seems funny to say "El papa le da muchas nalgadas al nino," at least in written speach. The "problem" in Spanish vs. English is that English has a specific verb: "He spanks his child a lot," but you would be less likely to say "He gives him a lot of spankings." Hmmm.
As noted above, the context would matter.

#3 wrote, "But if you are speaking about something habitual, it seems funny to say 'El papa le da muchas nalgadas al nino,' at least in written speach. The 'problem' in Spanish vs. English is that English has a specific verb: 'He spanks his child a lot,' but you would be less likely to say 'He gives him a lot of spankings.' "
Yes, exactly.

Are you trying to express the idea that the child is beaten by his father? You could use "golpear": "El hombre golpea a su hijo." It is less ambiguous - to hit/beat/punch.