They were late again, (much) 'to our annoyance'. [=it annoyed us that they were late again]
She shook her head 'in annoyance'. [=in a way that showed she was annoyed]
The long wait at the restaurant was a minor 'annoyance'. [='nuisance']
(..) in the highest levels of power. And it reaches down into the lives of every man, woman and child on this planet. So, of course, no-one believes me. I'm an annoyance to my superiors, a joke to my peers. They call me Spooky. Spooky Mulder, whose sister was abducted by aliens... (..)1998 The X Files