Squishy spot in lawn

I live in Central New York.  It has rained ALOT the last few weeks here and I have a few very troubling problems.  The house is about 18 months old.  The lot was all new growth trees (TALL anfd skinny with weak roots) when they cleared to build.  Very clay-ee soil, and slate underneath.

One area of my lawn has what we 'affectionately' call the pond.  It's basically a huge puddle, about 5 feet diameter and at times 7" deep.  It's right at the corner of the garage and I worry about the foundation.  Can anything be done aobut it?

I also have a squishy spot about 6 to 8 feet in diameter.  I can actually take a stick and poke it into the "spot" about 2 1/2 feet down.  Is there an aquifer or something down there and will I have a geyser if I dig it out?

The only solution I know of for this sort of thing inolves digging.

Either put in a drain trench or re-shape your lawn to be slightly taller near the house.