I'm sick and tired of doing the dishes. 나는 설거지하는 게 정말 지긋지긋해.

I'm sick and tired of doing the dishes.
나는 설거지하는 게 정말 지긋지긋해.
naneun seolgeojihaneun ge jeongmal jigeusjigeushae.

You can say
I’m sick and tired of the children I teach fighting one another

Or
It’s so exhausting teaching kids who fight all the time

Also your examples are also correct

There are many ways to phrase it actually

I’m sick and tired of mosquitoes

You only use a/an in a sentence
When the noun is singular. One thing or one person

Like for example 우유 2잔
2 glasses of milk( don’t use a/an here in English) because it’s plural more than one

우유 잔
One glass of milk( you can use a here when making a sentence)


I hope I didn’t Confuse you

You can also use a in certain situations like saying
A pack of wolfs or a group of kids

Also correct but people usually just use to first one because this one sounds odd even though it’s correct

You can say
I am sick and tired of this mosquito
Sounds more natural than
I am sick and tired of a mosquito