How can I repair Mr. Handy?

There is unfortunately no way to repair it when it's partially damaged ; you have to wait until he "dies", then you can "revive" him - for 2,000 caps.

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Disclaimer: the following might not be accurate anymore.

Here's a nice tip to optimize your Mr. Handy usage I found in this article :

When he gets near the end of his life, it might be a good idea to send him into the wastelands. Or even better, set him out in the Wastelands at night when you go to bed so he has several hours to gather caps. That way you can repair him ASAP.

You unfortunately can't heal him however I did read that you can revive him for 2000 caps

How to heal/revive and use Mr.Handy