Water Backing up Though hole in basement
You need to get the main drain line from your house to the city sewer cleared by a plumber. It'll cost you $75 to $150.
Here's how the drainage piping in a house typically is installed:
There is a big pipe that goes from just below your basement floor to just above your roof shingles called the "vent stack". All the drains in the house empty into the lower part of that stack and all the vent pipes in the house connect to the upper part of that stack below the roof to avoid having more roof penetrations than are necessary.
Once under the basement floor, that vent stack then turns almost horizontal and follows a shallow downward slope to the city sewer under the middle of the street your house is on. Typically, solids from your kitchen sink will accumulate in this near horizontal section closer to your house than to the sewer. Let's call that gradually downward sloping section of drain pipe the "Main Drain Line" from your house.
Buried around the bottom of your basement walls will be a set of drain pipes called the "weeping tiles". These weeping tiles allow excess ground water to drain away from your foundation, thereby eliminating the pressure causing water to flow through your basement walls. The water collected in the weeping tiles will either:
A) Flow to a sump pit in your basement floor, where it's pumped out somewhere by a sump pump, or
B) Flow to a "catch basin" in your basement floor. The catch basin simply looks like a drain hole in the floor. At the bottom of that drain hole will be a p-trap. Water flowing through that p-trap will then flow through a pipe which connects to your main drain line.
Now, what happens if the line from the catch basin connects to the main drain line upstream of where the main drain line is clogged with solids from your kitchen sink?
If there's a heavy rain, then water flows through the weeping tiles to the catch basin, and from there down the drain pipe to the main drain line. And if the main drain line is clogged, the water ends up backing up the catch basin onto your basement floor.
If your main drain line was clear, it wouldn't back up onto your basement floor cuz the main drain line could carry that water away as fast as it seeped into your weeping tiles.
As the main drain line becomes more and more clogged, you'll start to see your catch basin backing up when the clothes washer spins, or when you drain the bath tub, or even when you flush a toilet.
If you ever see toilet paper floating in that hole in your basement floor, that's a sure sign that your main drain line is clogging up.