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Normally blood is withdrawn from cuboital fossaa,where superficial vein
is there and no injury
to any other tissues occurs.
In your case may be ligament or nerve? may have come in contact of needle so some discomfort is there, usually this should go away with time and nothing serious occurs in such cases.
If it still hurting a local hot fomentation and/ simple paracetamol
tab may help to recover fast.
Avoid lifting something heavy for few days.
You will be OK soon