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If a PC’s max HP is temporarily reduced as a secondary effect when hit by a creature that deals necrotic damage, but then gains a level before a long rest, does their maximum HP increase in the normal hit die+Con mod way but still carrying the same sized temporary reduction in max HP?

For example: is it the case that a PC with a max HP of 13, temporarily reduced to 5 which gains a level increasing its max HP by 9 remains on 5 HP, but now the temporary max HP is at 14, rising to 22 after a long rest?

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It works exactly as you describe in your example

Each time you gain a level, you gain 1 additional Hit Die. Roll that Hit Die, add your Constitution modifier to the roll, and add the total (minimum of 1) to your hit point maximum. Alternatively, you can use the fixed value shown in your class entry, which is the average result of the die roll (rounded up).

Step by Step Characters - Class Features and Hit Dice

As written, current hit points do not increase when you gain a level; only your maximum does. You need to rest or heal to increase your current hp, even if you started with it equal to your maximum.

However, many tables play that increasing your maximum also increases your current by the same amount. At those tables, your example would have your current hp rise by 9 to 14.

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