The simulacrum spell description states the following:
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.
Up until the 2018 errata, many deemed simulacrums as being humanoids, and many had an agreement that healing methods (such as the cure wounds spell) did not work on them because allowing them to work would render the alchemical process above pointless.
But now that simulacrums are considered constructs, most healing methods do not work on them by default, since many of them have the following clause:
This spell has no effect on undead or constructs.
Now that easily available healing methods (that have the no-construct clause) no longer invalidate the alchemical process above, should the few healing methods that lack that clause – and are therefore construct-friendly (such as the regenerate spell) – be able to affect simulacrums? In other words, is that agreement still necessary?
Best Answer
Strict RAW: you can heal a simulacrum, but it goes against intent and common sense
There is no explicit statement in the spell description specifically forbidding healing the simulacrum via other means. Thus, by strict RAW it is not forbidden to do so. Just be aware that constructs are restricted from many healing spells.
However, allowing this goes against the intent of the spell as well as make the spell more powerful than it already is by providing lots more avenues for healing.
Rules as Intended and Common Sense: No, the alchemical process is the only way it can regain hit points
Rules as Intended
Jeremy Crawford makes the intent of the spell clear in a Tweet made before the 2018 errata:
(emphasis mine)
Rules as Common Sense:
Repair not heal
First note that this does not say heal it says repair. Though treated as a normal construct it is still a special being made of ice and snow. It is that reason that it likely has a special way to repair it.
Allowing other forms of healing means that the alchemical process would be useless
This statement must preclude other forms of regaining hit points otherwise it is meaningless. Even though it does not say so explicitly, the above method is clearly meant to be the only way to regain hit points for a simulacrum.
It does not say that potions or healing magic or rest doesn't work for example, but if they were possible then the expensive complicated process for healing would be meaningless and there would be no point in including it. Sleep especially makes this meaningless because it is free and available to every creature/class and allows for healing completely. Why would anybody ever pay 100gp per hit point if they could sleep or use any of the myriad of other methods for regaining hit points available? This logic also applies to traits or features or magic that let you heal.
The simulacrum being a construct doesn't change anything about the general case: they cannot be healed
The 2018 PHB errata has specified that the simulacrum is now considered to be a construct. As such, many healing spells will not even work on it explicitly. However, this does not change any of the logic above saying that simulacrum cannot be healed by means other than the one offered in the spell.