I always bring a "beach bag". I like to have it when we are by the pool. Sunscreen, phone (for pictures, sunglasses, reading glasses, something to read, wipes, etc. etc. Might have a hair clip/towel clip.
Evening - small purse - for tissues, reading glasses (sometimes), phone (for pictures), wipe/s (always have one or two even before COVID), hair clip.
During the day - it all depends what the plans are. Cooking class, Mixology (sp?), Bingo, shops, etc. I hav a small cross body bag.
There are not many times I don't have some sort of small bag with me. Unless running out for a minute and will hand over my phone to my DH. But many times we go in and out so might need sunglasses (can't be out without them more than a minute or two), might need reading glasses, always have phone handy for a picture, always need a tissue or two.
Not knowing what to expect for our first cruise, I brought some snacks. They were definitely not needed unless for your little one he might be a picky eater and/or have some special treat. You will always need wipes, water cup? but you are not too far away from anything.
I would pack something small, if you feel you don't need it, you can leave in the the cabin.