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Where these young trees or older specimens?
Older trees are difficult to successfully transplant.

Transplanting should take place in spring or fall and to avoid moving these trees during the hot part of the summer.
Make sure to plant the trees at the same depth, but dig the planting hole twice as wide as the root ball.
Mix 1 part sphagnum peat moss with 4 parts soil from the planting hole to backfill the tree. This will help with drainage and help to acidify the soil.
Water the trees throughly to keep the roots moist. When the first inch of soil dries, water again. Maintain the moisture level until the trees have become established.
The links below have more information and review the invasive qualities of this tree.

https://www.gardeningknowhow.com/ornamental/trees/mimosa/

This is what's known as a "reversion". Summer Chocolate Mimosa is likely a hybrid which has the special characteristic of chocolate colored leaves, but it still has the genetic potential to produce green leaves. The branches with the green leaves have "reverted" and begun expressing the green-leaf genes. If you want to restore the chocolate color, prune out all the green-leaf sections, since reversions will typically grow more vigorously than the rest of the plant.

If all the branches that produced leaves are gone, the tree is unlikely to make it. Mimosas are known for having frail wood and for often not living very long, so it might be better to cut the trunk down to prevent it from falling at some point. Mimosas can be propagated from its seeds, so you could plant a replacement if you have seed pods left.

This looks like a legume plant, perhaps a mimosa or locust tree seedling. If so, the seed that it grew from probably came in a pod and looked something like a bean (although many legume seeds are inedible or poisonous, so don't eat them!)

Mimosa do prefer a acid soil in the 4.4 to 5 ph range.
A soil test will help you determine if you need soil amendments.
Your County Extension service can help you with a soil test for a few dollars.
Here are some links to help you.
https://www.gardeningknowhow.com/extension-search/

https://www.gardeningknowhow.com/ornamental/trees/mimosa/mimosa-tree-weeds.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/soil-ph-plants.htm
https://www.gardeningknowhow.com/special/organic/the-importance-of-soil-for-an-organic-garden.htm

It looks as though your plant is just starting to form buds.

Give it time to start actively growing before fertilizing.

This link has more care information.

https://www.gardeningknowhow.com/ornamental/trees/mimosa/chocolate-mimosa-trees.htm