[RPG] If a simulacrum uses magic jar to control a new body can the new body regain hp by mundane healing

Reading the description in context gives:

You shape an illusory duplicate of one beast or humanoid that is within range for the entire casting time of the spell. The duplicate is a creature, partially real and formed from ice or snow, and it can take actions and otherwise be affected as a normal creature. It appears to be the same as the original, but it has half the creature’s hit point maximum and is formed without any equipment. Otherwise, the illusion uses all the statistics of the creature it duplicates.

What the simulacrum is is an illusory beast or humanoid that is partially real - illusions don't heal naturally or magically.

Reading the monster manual everything is a creature - constructs, undead, beasts, huminoids etc. Being a creature does not mean it can heal naturally or magically - some creatures can and some can't.

The simulacrum has the statistics of the creature and half its hit points - it does not have any of the racial or class features except for spell slots that cannot be regained so healing surges and short rests don't come into it.

If the simulacrum is damaged, you can repair it ...

You can repair it - you can't heal it.

Sure

Magic Jar provides opportunities for you to enter or leave a creature's body.

You can attempt to possess any humanoid within 100 feet of you [meaning, 100 feet of the container] that you can see (creatures warded by a protection from evil and good or magic circle spell can’t be possessed). The target must make a Charisma saving throw. On a failure, your soul moves into the target’s body, and the target’s soul becomes trapped in the container. On a success, the target resists your efforts to possess it, and you can’t attempt to possess it again for 24 hours...

While possessing a body, you can use your action to return from the host body to the container if it is within 100 feet of you, returning the host creature’s soul to its body.

If you wanted to do this quickly and often, you might be frustrated by the fact that the creature would be unavailable for 24 hours if it ever succeeded on the saving throw. And if the creature ever became unwilling, it would probably run away from the container once it had control of its body back. But otherwise, go ahead.

But watch yourself... literally

Note that Magic Jar lasts "until dispelled." And if the spell ends when you are more than 100 feet from your original body (or your original body is dead) you die. As such, if you are "sharing" another creatures body far away from your original one, and someone casts "dispel magic" upon the magical effect which is permitting your soul to occupy that body, then the Magic Jar spell could end. In that case, you would instantly die. In order to do this safely, you'd need to keep your body close by, and would also need to periodically feed it and keep it in relatively good health.

So take care with this strategy. As much fun as it could be to do an "All Of Me" style zany romp sharing one body back and forth, keep in mind you're messing with powerful necromantic forces here. If you're not careful, it could backfire with severe consequences.