Answer: Peonies do well in a rich soil that is evenly moist yet well drained. With a new plant your goal in watering is to keep the soil damp like a wrung out sponge, never sopping wet or bone dry. To know if you need to water, dig into the soil with your finger. If it is still damp, do not water yet.
When you do water, water slowly and thoroughly so it soaks down into the soil and encourages deep rooting. It is better to water deeply less often than sprinkle lighttly every day. After watering, wait about six hours or so and then dig down and see how far the water went; it can be surprising.
You do not want to over or under water. Sometimes with a clay soil you can overwater by accident, and the clay will hold the extra water in the bottom of the planting hole and flood the roots. This can cause root damage which then can cause wilting; but, underwatering can also cause wilting.
Sometimes a new plant is still developing its root system and may wilt a bit in the afternoon but then perk up by evening. When this happens, it is already watered enough it is just having heat stress. If it is wilted the next morning, it probably needs more water.
As far as fertilizing goes, you can top dress in early spring with a good quality compost (home made using a wide variety of organic materials is great) and/or thoroughly rotted stable manure and bedding, and/or apply a complete fertilizer such as 10-10-10 in either slow release or regular granular form. Read and follow the label instructions. Maintaining a layer of organic mulch several inches thick year round also helps feed the soil slowly as it breaks down.
For a more detailed fertilizer recommendation you would need to run some basic soil tests and then use the test results to identify any specific nutrients that might be lacking. Your county extension should be able to help you with testing and interpreting the results. Enjoy your new peonies!
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