can I apply again after refusal?

Ok this is the full story. Student visa expired since october 2012, home office still has my passport and seems to have lost it. I am now living with my british unmarried partner and our british child. She is currently at home caring full time for our daughter. I cant work because I have no valid visa. We applied for family life as a parent on the 10 year route.

I applied for leave to remain as the Parent of a British child. They have turned me down because I didn't provide DNA evidence to prove that my daughter is mine. I couldn't afford the DNA test as I can't work. But they refused to exempt me from the DNA test fees as I requested, and refused my application instead. They claimed I Refused to provide information to support my application. They have refused me right of appeal too. And told me to leave the country because I am an overstayer.

I want to know if I can make a new application again as an overstayer. I made my first application as an overstayer anyway. So can I make a new application, this time with a DNA test result proving paternity of my daughter?‎

Duplicate post.