Bog garden.

I agree w/ Izzy & think if you dig out and put in some good soil, you'll need no special drainage. In a rain, a certain amount of water will run off onto walk and another amount will sink in and keep on sinking ... like all the other parts of the lawn.

Also, I would remind you to put down a tarp over that new cement because any red clay dropped there could stain it easily. Good luck with it.

Depends on your local soil. Here in VA with our heavy clay, that'll be a swamp without some way for water to exit. Digging down a foot and replacing the soil will just put the level of your swamp a foot lower. If you live somewhere with decently draining soil, though, digging it out could certainly help.

It’s a lake for 2 reasons: water is getting in, and water can’t get out.

The first thing I would do is, like others said, dig out some clay and then fill the bed with good dirt, just below the level of the sidewalk. Then assess.

You might possibly need to insert a drainage pipe from the bed, under the sidewalk, so excess water can drain away from the house.