Nectar

[Lizzie]

Hello, lyon. Welcome to the site. A couple of comments for you......

I beg a -- I second Todd's point that some of the line breaks are not being well utilized. There's no reason that "a" needs to be the standout word in this line.
mosquito
who refuses,
despite the many times 
I throw her off:
Let me alone! -- this stanza reads long and clunky. I'd re-word.

but mosquito will not. -- THE mosquito?
She simply waits -- I also like the gender specificity of the mosquito -- makes me immediately think "relationship poem."
when displaced, -- a stronger word choice than "displaced," surely! Evicted, rebuffed, etc.....make every word pull its weight.
for a still 
and fresh spot
to suck.

Relentless! Let me 
alone! I am angry -- I don't feel the benefit of this emotional statement. It can be deduced from "thrashes." You could try another image that would demonstrate anger instead of coming right out and saying it.
but mosquito ignores -- THE mosquito?
my thrashes, again -- there's a grammatical situation in here....maybe say, 'the mosquito again ignores my thrashes, finding another spot to suck.' 
finds another spot to suck -- I like "suck" and "suckle" not only for the sonics, but because there's sexual and dependency connotations, respectively. Makes me think of a relationship where one person is a parasite.

and suckle.
Faith-full mosquito… -- I like the specificity offered by putting the dash in between faithful

you always find me useful. -- Good ending. No question now that the poem is not about a mosquito.

Overall, I'd say that you need to get the grammar and punctuation nailed, for sure. That's a gimmie.

Second, I'd work on your line breaks so that each word at the end of a line is something your really want to emphasize.

Third, I'd make sure that every word is as strong as it can possibly be. I'd make it as tight and concise as you can.

I look forward to seeing where you choose to take this one. Hope this helps!

Lizzie