If it's soggy, you have to deal with that first. That's a drain. A drain pipe in the lowest area covered by gravel is probably your cheapest option. Plant ferns to the sides to make it attractive.

Please add a photo showing the front of your house so we can see the style of your landscaping. You can add photos in the comments. Thanks.

Landscaping is bare minimum builder grade.

You didn't mention budget, but this is what I'd do. If it would handle the runoff, I would probably do a swale or french drain down the side and at the bottom, then create a dry creek bed over the top on the slope. Integrate that into steps or terraces down the side. Plant shade loving plants, terrace the planting area on the right, if needed to keep the water from running down that area and keep the plants in place.