No, you can't pay for placement. Please search the forum for advice on promoting your games. People ask all the time.
I have looked through some of those posts, and already, as I said, use several streams of advertising, to maximize a return on my investment. But you did answer the question I saw on none of those posts, which is that I cannot pay to be bumped here. That leaves me wondering why my games do not show up when searching for tags I use on them, but occasionally do with other tags. Or sometimes show up with a tag, but the same search another time, they do not. Have any idea about those things?
Actual search is capped at abou 60 results. You need to select tags in the tag search box. I stumbled about this too. It read Search for games, but it should read Search for game titles. Yeah, yeah, it noe also shows search for tags, but this only adds to the confusion. It does search for suggested tags. It does not search for game by tags. So if you searh https://itch.io/search?type=games&q=turn-based it will show you the related tags
Turn-based
Turn-Based Combat
Turn-based Strategy
Card Game
plus games that have this in their title.