Can I Bring Less Money to Closing of Fannie SFH

[Brad D.]

Doing my first deal, set to close January 25. In the future I'll get more creative, but going the safe route at first: Fannie Mae Investor Loan on SFH, 10% down, doing deal through mortgage broker reccommended by Real Estate agent. Got the house for 31,000, and house should rent for 650-675. Here are the details of the transaction:
Purchase Price: $31,000
Estimated prepaid Items: $1,081.40
Estimated Closing Costs: $1,916.62
Total Costs: $33,998.02

Other Credits
Cash Deposit on Sales contract: $1,000.00

Loan Amount: $27,900.00

Cash from Borrower: $5,098.02

So, I put down $1,000.00 earnest money, and they're saying I'll have to bring $5,098.02 to close. So it's 6k out of pocket to get this 31k house. I'm wondering if there is a way to wrap up more of these cost into loan and bring less cash to deal in a fannie/freddie type situation? If so, is it too late to do it on this deal?