If you could untill the ground and remove the peat you'd be ahead of the game. Where do you live? Is the area shaded or sunny most of the day? We need that info to help you more. Your zone doesn't matter, we need the location.

Most people who think they have clay have other stuff going on. If they try to fix it like it was clay then they are doing the wrong thing. There is a test called the jar test which can help you decide if you need further testing. Get a straight sided jar, fill it half way with soil, and measure the amount of soil with a ruler next to the glass (TAKE A PICTURE). Fill the jar with water and a short squirt of shampoo or dish soap. Cap the jar and shake it until all the soil is dissolved. Set the jar down, hold the ruler up and take another picture. After 2 minutes take another picture with the ruler. After 2 hours take another picture. After 2 days take another picture. What you will see is settling of sand/gravel (2 minutes), silt (2 hours), and clay (2 days). If you can see through the water on top of the soil after 2 days you don't have very much clay. Clay particles remain suspended for days and days to cloud the water. You can divide the sand by the total to get percentage of sand. Subtract the sand from the 2-hour picture settling and divide by the total to get the silt percentage. The rest will be clay. When you discover that you don't have any clay at all, then you might want to get a soil chemistry test to see what is the problem. Likely it is an imbalance of salts in the soil with too much magnesium being a possible problem.

It gets some sun, about 4 hours direct a day

dchall and I disagree on this point but clay-based soils are extremely common!! And it doesn't take 2 days of the jar soil texture test to confirm that!! Just shake the jar well and let it sit overnight - the layers will be very obviously delineated the next morning - sand at the base, then the silt and finally the clay. Anything floating or clouding the water at that point will be organic matter, not the clay. I teach classes on soils and the jar test is one that I ask all my students to perform. In very few cases does the test confirm no clay. It tends most often to be the primary component of the 3 different soil types,

However all that aside, I do agree that adding peat is not what you want to do to successfully amend a heavy or poorly draining clay soil. The particle size of the peat is far too small to increase porosity (which is what results in better drainage), peat is far too water retentive to be an assistance in drainage and it will acidify the soil, at least initially. And if the peat was not adequately moistened prior to any incorporation, it will be hydrophobic and repel water. Typically the suggestion is to incorporate a quantity of coarse organic matter, like compost.

Okay so...now what??? Here's what I would do to try and get moisture down into the soil as amended. I would put a few ounces each of molasses and any clear shampoo into a hose end sprayer and fill the sprayer bottle with water. Then spray that all out on your little piece of garden. The foamy sugar water should help "wet" the soil surface and allow moisture down inside. You may need to continue to turn the soil as you spray. Both sugar and soap tend to allow moisture to penetrate more easily. This might be a saturation project to get the soil to take the water. Once that dries a little, then you can seed into it.

Having said that, seeding in the spring is usually a poor idea because crabgrass seed is germinating at the same time. If you do seed now, use an inexpensive seed and plan to redo the area in late summer. Write back to tell us where you live and we can be more specific about seed types and timing.