Yeah, I’m not sure pesto and white wine are the easiest match.
I often add cream cheese in with pesto and a glug of pasta water, and / or a splash of milk, to make a creamy sauce. I’ll use cream cheese or creme fraiche depending on what’s in the fridge. Sometimes both! I’ll fry up some onion, mushroom, leeks etc, or add peas in, add the cream cheese and pesto, warm through, add pasta water and cooked pasta.