Consider buying as much house as you can now with the $20k down, then using the gift money to make some improvements or to it apply it to principle, when you get it.
There is a lot of inventory available now, so you should be able to find a home that suits your needs and your wants, that is within you budget. Good Luck on the search.
Just a thought.
Or.........see what you qualify for currently and get the ball rolling with a lender. Get PRE-APPROVED, not pre-qualified. The lender can always up your purchase power after getting the actual money for the down payment.