There are several synonyms for the phrase "annoying pest." For example, an irritating pest might be called an aggravating or vexing pest. Similarly, a bothersome pest could also be referred to as a pestering or pestiferous pest. If the pest in question is more persistent and difficult to get rid of, it may be termed a persistent or tenacious pest. Furthermore, if the pest is particularly loathsome or repugnant, one might call it a disgusting or repulsive pest. Overall, there are plenty of ways to describe an annoying pest, depending on the specific characteristics that are irksome.