What's an adverb meaning "focused on one thing/goal"?

Single-mindedly fits the bill quite nicely, I think...

Single-minded:

having one driving purpose or resolve : DETERMINED, DEDICATED
_{from m-w.com}

having or showing a single aim or purpose:
a single-minded program.
_{from dictionary.com}

Plus, of course, the suffix -ly, "a suffix forming adverbs from adjectives: gladly; gradually; secondly." (also from dictionary.com)

Or, as Cambridge has it directly:

in a way that is very determined and thinks only about achieving a particular thing
_{from dictionary.cambridge.org}

(I would argue that their definition would be improved by saying "... achieving one particular thing".)

"The American Federation of Labor relentlessly campaigned for higher wages throughout its existence."

Meaning of relentless in English ... US /rɪˈlent·ləs/ continuing in a determined way without any interruption:

https://dictionary.cambridge.org/us/dictionary/english/relentless