As written, yes, but that is not the intent
The text of the simulacrum spell says:
The simulacrum lacks the ability to learn or become more powerful, so it never increases its level or other abilities, nor can it regain expended spell slots.
If the simulacrum is damaged, you can repair it in an alchemical laboratory, using rare herbs and minerals worth 100 gp per hit point it regains.
You could maybe argue that regaining hit points by using second wind makes the simulacrum "more powerful" than it was when it had fewer hit points, but that is a pretty stretched interpretation of that line, because it also loses the ability to use second wind, and that makes it less powerful by pretty much the same measure. And apart from that line, there is no text that forbids your healing the simulacrum in any way.
The text says that you can repair it, but it does not say that you cannot heal it. It does not say that the only way to restore hit points is to repair it. And in 5e, constructs can be healed like other creatures just fine, unless they have text saying otherwise. So, technically, as written, you can heal it, and probably also use second wind to heal it.
You also can argue, as this answer does, that the alchemical procedure would be meaningless if there were other significantly cheaper and simpler repeatable methods of healing the simulacrum. Nobody would spend all that time and money and hassle, if they can just lay on hands or cure wounds their simulacrum. And as you can see from the linked questions (and the highly upvoted answer on this one), community consensus is that is how it should be played. As you will see below, it is also how the designers intended it. But it is not how it is written.
Designer intent: Regaining hit points
There are multiple tweets by Jeremy Crawford, that the alchemical procedure is the only way to have the simulacrum regain hit points, and this is not a case where he is inconsistent. There is this tweet:
Q: Can the creature created by Simulacrum be healed by spells?
A: To restore hit points to a simulacrum, you must use the costly alchemical procedure mentioned in the spell.
He does not make any exception, not even for Wish. Gandalfmeansme found yet another tweet asking about hit dice explicitly where Jeremy again confirms:
Q: Is the intent of Simulacrum that it can be healed through magic,hit die/resting,or ONLY through the alchemical process?
A: The intent is that the simulacrum (PH, 276) can be healed only by the alchemical process.
Designer intent: use features once and then nevermore
Additional tweets by Jeremy state that a simulacrum cannot regain spent features.
The creation is meant to be unable to regain use of any of its features that it expends (RAI)
In a followup he suggested there may be errata to this effect in the future and that
A simulacrum is meant to lose efficacy over time, essentially running out of juice, until only at-will abilities remain.
You could argue that regaining hit dice is such a not-at-will ability, and thereby it should be possible to use the hit dice the simulacrum got during its creation at least once, during a short or long rest.
However, since specific overrules general, then the specific, explicit ruling about hit points overrides the more general one about using up features
Conclusion
Jeremy Crawford's tweets are not official any more. You can of course decide to ignore his missives about designer intent, and rely only on what is written in the official rules.
Consider that simulacrum is widely seen as a spell that is already broken, effectively doubling your spell slots and actions and providing unlimited ritual casting. I think it is wise to employ any reasonable restrictions.
You should clarify how this is handled with your DM.