"He cannot take good care of himself."
Translation:Hij kan niet goed voor zichzelf zorgen.
You are using the word order of a reflexive verb, but this isn't one. "Zorgen voor" = to take of. You definitely need the preposition. The adverb "niet goed" must come before the prepositional phrase ("voor zichzelf").
Edit: you could also use the transitive verb "verzorgen". In that case, it would become: "Hij kan zichzelf niet goed verzorgen."