Can I add a moisturiser after Adapalene?

Yes.

Adapalene (aka Differin) is usually better tolerated than some other retinoids, but it can still make skin quite dry and irritated for a few weeks.

Just ensure that you choose a moisturiser that doesn’t contain AHAs, PHAs, Urea or Vitamin A - anything that might increase the penetration of other ingredients in the moisturiser. Adapalene doesn’t seem to be absorbed into plasma through topical use, or not much anyway, but it will still tend to leave your skin less protected, and you don’t want to exacerbate this.

So I would choose something fairly light and straightforward, with a water content above 75%. Look for some of the following ingredients: aloe vera, seaweed extract, oat protein, calendula extract, chamomile extract, borage, mixed with gentle oils and butters: coconut, hemp seed, cocoa butter, andiroba avocado, argan, rosehip, kukui. Until your skin is feeling hydrated and any redness has gone. (You don’t need to find something with all those ingredients - aloe vera alone can be great - just make sure you’re buying aloe vera and not a jar of glycerine with additives!). Hydrosols would also be a good idea, in terms of moisturising without irritating. They are not oily, but have enough to hydrate skin. I would recommend: rose, seaweed, chamomile, camellia, cornflower, calendula, honeysuckle and possibly lavender.

For now, stay away from moisturisers with active ingredients that could irritate, including Vitamin C - even though this can be healing, I wouldn’t apply it after this sort of treatment if the skin is inflamed or excessively dry.