If you are adding a EPDM liner you can put the liner over the hole floating it on top of the current water. Then add water on the liner and the weight will push water out of the hole as you fill the liner. This is tricky and adjusting liner position is very limited. And this only works if water being pushed out of the hole can flow away. Any pond where ground water can't escape when pushed out has a risk of floating so I would address that too if an issue.
The water in the hole will likely go away at some point via soaking into the ground and evaporation...unless more rain comes. However, when water soaks into the compressed dirt walls the dirt becomes soft, expands if clay, and in many cases the wall will collapse to some degree. Sometimes the walls are undercut causing problems. So your 4' deep pond becomes 3' or 2' deep. Digging out mud is even less fun than packed clay. And the result is a larger pond because the wall(s) moved out. If the liner has already been purchased it may now be too small.
Most builders pump it out as mentioned. But they do it quickly, before the clay walls are saturated too much.
An EPDM liner can be used as a tarp, because it will be sized to be much larger than the hole opening, by using some lumber to span the hole with a good slope. Has to be secured well though against wind and water can't collect in the "tarp" or it will pull the tarp into the hole. A billboard tarp can also be purchased for this for pretty cheap.
Once the water is gone the next problem will be the mud in the bottom. If you set the liner on the mud you may not be able to walk on top of the liner without sinking into the mud pushing the liner into the mud. This makes setting the liner almost impossible as it sticks to the mud. You may have to wait until about August when the mud dries enough assuming you found some way to cover the hole to keep out additional rain. Laying old carpet on the mud can be a solution even if a couple layers are needed.
This all assumes an EPDM liner. Poly liners aren't as tough.
