Nectar

[lyon]

Hi Lyon, Let me give you some comments. A lot of them will probably be style choices more than anything else--so largely subjective. 

I beg a--I am not a fan of this first line. The break on "a" doesn't seem especially strong. I would be tempted to move a down to the next line and end on beg. It seems more in keeping with the negotiation one would do with a mosquito.
mosquito
who refuses,
despite the many times 
I throw her off:--The fact that it is a her (any gender at all actually) gives the mosquito a figurative sense. It is a potential stand in for a relationship.
Let me alone!--I think this line and where it repeats can be cut without impacting anything in the poem. 

but mosquito will not.
She simply waits
when displaced,--You could cut "when" potentially.
for a still 
and fresh spot--You could cut "and" also.
to suck.

Relentless! Let me--I don't think relentless gives you much it is a tag explaining action which pulls us out of the moment. 
alone! I am angry
but mosquito ignores--perhaps a slight refocus start the strophe with "mosquito ignores/my anger/my thrashes, again 
my thrashes, again --Nice break on again.
finds another spot to suck--Potentially you could emphasize your earlier choice and break this line on spot allowing to suck to sit alone on its own line. 

and suckle.
Faith-full mosquito…

you always finds me useful.--probably want find here. 

I hope the comments help some.

Best,

Todd

Thank you for the close read and comments! 
I'm wondering how you felt about the beginning and overall pace of the poem? Did it deliver the ending too quickly?
Thank you so much again!