Jisho

The verb つとめる can mean "to work." It's important to note, though, that the entry linked above is for it as an adverb.

Regarding 仕事, it's a noun that can be turned into a verb with する.

But つとめて would typically be the て form of the verb, or an adverb as linked above.

Another thing to keep in mind is that つとめる can be written with several kanji, and not all of those nuances are related to something similar to 仕事.