Rock Star Ate My Hamster

Not heard of it, only Freddie Starr..
Bottom left looks like Russell Brand

I had this… Good old Codemasters and their pocket money games (edited)

I believe it was a 1980s thing. Plenty of emulators, a few loosely similar games from back in the day - Minder springs to mind. Sensible Soccer has a similar style game play.
I believe the genre is referred to as a "management" game.
There are plenty of management style games available on the Android platform, none quite as quirky as this entry though.
Dungeon Keeper was a fun game - worth a look?

For your viewing pleasure
archive.org/det…28K

Loved this game. I also had it on Spectrum