gentle facewash, similar to simple moisturizing face wash? and routine for when its really hot?

Hey everyone. I'm nearing the end of my "simple" moisturizing face wash and I'd like to get something korean ideally. My skin is usually dry (except during that time of the month). Also, last week we had a really hot week and my face completely broke out, enlarged pore, black heads, white heads, everything.

I usually just use face wash (only sometimes, usually near that time of the month), then some hyaluronic acid toner (torriden) and then BHA, then torriden dive moisturizer.

It doesnt get very hot often in this country (UK) so I really dont know what I should be doing. I realise I need to be more on it with facewash, especially when it's hot, so I look forward to advices and suggestions.

I'm someone who doesnt wear make up and barely know anything about skincare so please guide me.