What's your reason for keeping it up now?
I had a copyright claim for this trailer analysis. as you can probably guess, the trailer itself wasn't 18 minutes long lol! It was me reacting, and then analysing the trailer frame by frame to study it so I considered it transformative enough. I didn't bother to appeal it because there's always the chance that appealing a copyright claim can lead to a copyright STRIKE if the original creator (in this case Universal Pictures) decides that they don't want to let go of the copyright claim.
I kept mine up because it wasn't a strike, it still fit in with the theme of my channel, and I consider it educational. Overall it hasn't harmed my channel at all, so I don't mind losing monetisation on that particular video because it was a one off occasion.
In terms of recording footage and then realising there's radio music playing in the background, you can either try to edit out the music but cutting a portion, if you're not talking, you can replace the music, but if it's on YouTube already and you got a copyright claim, there's nothing you can do outside of using YouTube's editor to either edit out that portion, mute it, or add some music from the audio library. If you're talking while there's copyrighted music, you'll just have to accept that you're going to lose that bit of verbal track. If the entire video or a good portion of the video has copyrighted audio in it, and editing it in YouTube means you'll lose too much dialogue, then there's not much point in keeping it up on your channel.
If this is BEFORE uploading to YouTube and you realise that there's a radio in the background, depending on how loud it is, you may be able to filter out the audio using an audio filtering program. It may make you sound a little robotic but I've used it at times to try and remove some background music. Not always great results, but better than nothing.
I did this for a video somewhat recently.