Although there are plugins out there that can clean up background noise (Waves NS1, izotope RX8, Sonnox DeNoiser), by far the best method is the manual approach, (definitely forget Logic’s Denoiser for this task)!
You can use “Strip Silence” (destructively) in the file editor, (or gain reduction), but by far the best approach is to work in the main window, (non destructively). Zoom into a vocal region & simply cut out the noisy portions. You could use the marquee tool for this, but my preferred method is scissors - choose your 1st tool as pointer, 2nd tool as scissors. (When you hold down cmd, the 2nd tool is active). It might take a while, but once you get going… it’s surprisingly quick.
Remember to keep the breaths in, & you will definitely need to add fades - usually tailored for each edit, (curves can help a lot with this). Sometimes you might have a breath with noise underneath - in which case you’ll need to copy a breath from a quiet passage, & splice it in. Longer fades & edits usually work better than abrupt ones, (& it’s a good idea to solo your vocal track to hear what’s going on). If you can get a noisy vocal sounding half decent when solo’ed it should be OK in the mix…
Compression raises the noise floor, so if you’ve already got a noisy background, compression will accentuate this, so try easing back on compression, (or rely on volume automation more than compression where appropriate).
Best of luck!