Is it possible to finance a home with average credit, an average paying job I just started and $40k of debt?

Depends on where you live...look for a lease to own deal where some of your rent is put toward the down payment. This is a buyers market right now. Sellers will be more willing to negotiate especially if they are anxious to sell. Source(s): Experience in real estate & ownership

I'm sure you can find a mortgage company to finance a home for you, but it might be beneficial to get your debt down before trying to buy a house. It might also be harder to get financing since you just started a job therefore you wouldn't have the job history that sometimes they look for.
If you do get financing it will probably be at an interest rate much higher than someone with good, established credit. Look into buying an equity and assuming a loan. Have did this several times and have also sold a couple of houses this way.
When defining "average" income, how does that compare to your income? For example, if you are making $30,000/year, you may have a much more difficult time getting a mortgage due to the $40k debt.

Pay down as much of your debt that you can. Get a second job if you have to. That way, your debt decreases and at the same time, your credit rating will increase. Two birds one stone kinda thing.