"This man is writing a letter."
Translation:Ten mężczyzna pisze list.
Perhaps this link can help: https://www.clozemaster.com/blog/polish-demonstrative-pronouns/
Your sentence is correct, but it doesn't mean 'writing a letter', but 'writing a list' (a shopping list, to-do list, etc.). It's the Accusative form of "lista".
For "list", the Accusative form is also "list". Accusative is identical to Nominative for masculine inanimate nouns. Same goes for all neuter nouns, and for 'not masculine-personal plural' nouns.
"mężczyzna" is an adult male person and "człowiek" technically means "a representative of the homo sapiens species". However, when you refer to a specific adult male person, actually "człowiek" often feels more natural.
For example here, I'd indeed be more likely to use "Ten człowiek".
If we were to say "John is a good man/person", that's also rather "John jest dobrym człowiekiem" then "John jest dobrym mężczyzną", which sounds like "John is a good adult male".
In a different context, e.g. translating "I am a man", on the other hand, "człowiek" and its forms would sound like the name of the species. "Jestem mężczyzną" means that "I am a man" (I am male) and that's what I'd say. "Jestem człowiekiem" sounds like "I am a homo sapiens" and you'd say it in a context of fantasy or science-fiction.