Offer arrived, can I still ask for more? - 10 months and counting

It is the first offer in 10 months and I do not want to mess it up. I have read a lot of threads here and could not find my exact situation. I have a competing interview next week. Did it backfire?

Most employers expect a short conversation. Mine did not even blink.

I thanked them and said I was excited but hoped we could talk about the pay. They came back with a bit more.