Nectar

[Todd]

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