While the term ‘annoyment’ is not commonly found in standard dictionaries, it is sometimes used informally to express the state or action of being annoyed. The proper noun for the feeling might be ‘annoyance’, which is widely accepted and understood in the English language.
‘Annoyance’ refers to the feeling of discomfort or irritation caused by something bothersome. On the other hand, ‘annoyment’ may appear in casual conversation or writing but is not recognized as a formally accepted word in most English lexicons. So, if you’re looking to communicate clearly and effectively, it would be better to use ‘annoyance’ instead of ‘annoyment’.