Is it possible to power a 4K display at 60 Hz with a DisplayPort/HDMI adapter connected to a USB-C hub?
The 2018 MacBook Air actually has 2 Thunderbolt 3 ports. I.e. they not just USB-C ports.

This mean that you can definitely connect all sorts of multiport adapters, docks, etc. to enable you to charge the Mac while connected to a DisplayPort monitor - and then even more.

Just as an example you can use the OWC Thunderbolt 3 dock and connect that to one of the two Thunderbolt 3 ports on your MacBook Air. It comes with a power brick that will charge your MacBook Air through that single cable. In addition it has a 4k at 60Hz DisplayPort output (you'll need a mini-DisplayPort to DisplayPort cable). 

This leaves you with one of the two ports on your MacBook Air unused - as well as providing lots of extra ports on the dock itself (5 x USB 3, 1 x USB-C, 1 x daisy-chained Thunderbolt 3, ethernet, S/PDIF, audio-jack, SD-card and micro-SD).

In fact you could also use the daisy-chained Thunderbolt 3 port to hook up the monitor - then you would require a USB-C to DisplayPort cable instead.