Suddenly saw a small blood clot in my eye. I have chest infection. Hasn't recovered. Due to blowing of nose? That was the last thing I did. Help

Probably, yes. If you're describing an area of blood overlying what would normally be the white of your eye (the "sclera"), you're describing a "subconjunctival hemorrhage" - a completely harmless condition which will go away on its own and needs no treatment. It can often happen after straining, as with blowing your nose. If the blood clot is truly *in* your eye - in front of the colored part - go see an md.
Hemorrage in eye. Yes sometimes when you forcefully blow your nose or cough a lot you can break a superficial blood vessel underneath the conjunctiva, called subconjunctival hemorrhage.It is self limiting and with time will disappear.