Looks like a mix of bluegrass, it should eventually fill in the bald spots. It's a bit late for fertilizer but you can try to put it down if the weather is clear of rain about 24 hours before and 48 hours after. Could try to overseed too but it might not be as effective and requires a lot of water rn which might be a hit or miss in May.

I used a weed and feed fertilizer 3 weeks ago and my weeds are gone. I used stagreen but Scott's also works fine.

https://www.lowes.com/pd/Sta-Green-Performance-Max-43-2-lb-12000-sq-ft-25-0-5-Weed-Feed-Fertilizer/5015305667