Can Mr. Handy's exploration outside the vault be upgraded?

Solution 1:

No, Mr. Handy cannot be upgraded. He doesn't gain experience points and cannot be healed by stampacks.

About the loot, from experience, he will collect more and faster if you let him a few hours in the wasteland. He is very rare in lunchboxes but he can be bought for 1 euro (or around 4 euros for 5 Mr. Handy). It's not a bad price considering you can still revive Mr. Handy after he dies for 2000 caps. So you can keep him forever :) (but cannot be changed between vaults)

You can learn more about Mr. Handy here.

Solution 2:

Yes, there is a way. Simply wait, he'll find better and better stashes of Caps the more you leave him in the Wasteland.
Example:in the first few hours / days, he will find a minimum of 1 or 2 Caps. In the next Days, you will see a minimum of 3, then 5, etc.