I'm currently reading the Engines of Light trilogy by Ken MacLeod. In that universe, there is no FTL travel nor communication, but starships can travel literally at the speed of light. There are "spheres of influence" spanning some tens of lightyears, and the information delay between worlds is indeed a plot element.

Many collonial empires worked with such latency before trains, steam ships, telegraph and wireless comunication became available.

Vernor Vinge predicted a future galactic usenet - unreliable messages that might take a while to arrive, or need to go via multiple possible paths, are what usenet was built for.

The communication delay is one of the key plot elements in The Three Body Problem books.

It's a major plot element in Forever War.