There are several options that may work for you, but your first step would be to speak with a lender. As Tamara mentioned, homes in the Antelope Valley and surrounding areas are very affordable right now.
Below is the contact information for a reputable lender. Feel free to call or email him - he can tell you whether or not you qualify for a loan, and if so, what your payments would be based on the price you are qualified for. He can also guide you in credit repair if there are any errors on your credit report that need to be removed.
661 206-6345 Office
661 406-5629 Cell
661 206-6382 Fax
Lancaster, Ca. 93534
NMLS ID: 260877
DRE ID: 01299857
Hope this information is useful to you.
Keller Williams Realty
1401 West Rancho Vista Blvd #B
Palmdale, CA 93551
A couple things to look into:
1 buying a mobile home cash in a park w/space rent and utilities at about $600/month, possible in the Antelope Valley
2 using the $20k for a down payment and keeping your mtg PMT at abt $600/mo, possible buying a City of Palmdale home and utilizing their down payment assistance
Before doing step 2 you'd need to sit down with a local lender and discuss your situation to see if we can get you qualified for a loan.
Let me know if I can help you with either of the above,