When an undead character gains Corruption, it reduces their current and maximum hit points. But temporary Corruption works the same as it does for everyone else. So if the undead character has temporary Corruption and takes a short rest, their temporary Corruption (if any) goes down by their proficiency bonus automatically.
Let's imagine that you are making an undead mystic with the Sorcerer approach. At 1st level, you have a Charisma modifier of +3 and a proficiency bonus of +2. This means your Corruption Threshold is 10. You have a Constitution modifier of +2, thanks to your origin's ability score increases. You have 10 hit points.
You can select two cantrips and one of your 1st level spells to be favored (favored cantrips are 0 Corruption and the chosen spell is now only 1 temporary Corruption). That means there are two spells that won't hurt you at all, and one more powerful spell that you can cast twice without long term consequences as long as you can take a short rest. And, until you can take that short rest, you are at 8 hit points instead of 10, so you need to be careful, but it's manageable.
When you take a long rest, you reduce your temporary Corruption by twice your proficiency bonus automatically.
Let's say are undead Sorcerer is now 3rd level. Their Corruption Threshold is still 10, but they now have 24 hit points.
They will take a ritual spell at 2nd level, giving them 1 permanent Corruption. So their hit point total is now 23.
They still have that favored 1st level spell, but now can have a favored 2nd level spell, and four favored cantrips. So that's more options: four spells that don't cost them anything, a spell they can cast twice with a short rest, a spell they can cast once with a short rest, one spell they can cast as a ritual, and if they need to cast their unfavored 1st level spell, it is only 1d4 temporary Corruption (due to The Shadowed Slope and their ritual spell), so a long rest will allow them to reset after that. And, worst-case, they go from 23 hp to 19.
And, of course, extended rests reset all temporary Corruption automatically and restores all hit points, so an undead party member is going to be pushing hard for those opportunities.
In short, playing an undead mystic is hard. Very hard. But, with very careful selection, you can make it work. And, when you get to higher levels, you can take on a fair bit of temporary Corruption without it affecting you too much (any mystic should be as far away from the frontlines as possible, of course) when you do need to lay down the law.
In return, you're immortal, don't sleep, don't need to eat or drink in most cases, are immune to poison damage, diseases, and the poisoned condition. There are trade-offs, and it's up to you if you want to try to make it work for you.
As for healing via items, take a look at Holy Water. Elixir of Life will heal more (and should provide a net positive if you took actual damage), but Holy Water will heal some and reduce temporary Corruption (essentially "healing" an undead character even more). So an undead mystic needs to have a friend in the church.
