It might be possible to disable by changing your notification settings.
Pop-ups for training and puzzles

You can try that setting option mentioned above;
https://www.chess.com/settings/notifications
Alternatively you can get Ad Blocker for your internet browser. I use Google Chrome with ABP (Ad Blocker Plus)

I don't think it's an ad though? Maybe an ad blocker would block it!
I have Leagues and Streaks off (now). Maybe it's hidden in there, although it isn't really either. Let's see.
I've started to get a little pop-up every time(?) I log in, at least on desktop, telling me I could do some training or play some puzzles. I really don't want to be badgered in this way, and I can't find a setting to disable the pop-ups? Please make it stop, or it would be great if anyone knows how I can do that. Thanks.
(I'm sure the site's data tells chess.com that these pop-ups drive X number of people to puzzles or whatever, but it won't tell them how many people roll their eyes and get annoyed before clicking away...!)