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This is unlikely as it's just an online game rather than a PC game, but just in case you've confused any of the details, it sounds vaguely like Adventure Elf. http://www.primarygames.com/holidays/christmas/games/adventureelf/
Nope, that's not it, I already had seen that and that's not the game. There was no timer and I'm pretty sure the plot didn't revolve around penguins stealing gifts. And the game was strictly a PC game; an internet connection was a luxury for me back when I was little (as in it wasn't very affordable and common), and so there was no way this game could've been a browser game.
After years and accidentally remembering this again but struggling to remember the name, I hereby bump this once more.
It would appear that I never mentioned there being one particular level where there were tons of candy canes floating about. Collecting them would allow you to go all over the level since your "fuel tank" would get so big from all the candy canes. Besides that though, I don't recall anything else I'm afraid.
^ I wish — it isn't sadly. I remember the graphics being more detailed, and the main character wasn't Santa. Entire levels didn't fit on one screen iirc. I'm also fairly sure the game supported mice, there was a level select screen that I think was arranged like one of those advent calendars with the last level somewhere in the center and I think the baclground of that one was something with a decorated Christmas tree, perhaps in a snowscape.

I've been trying to figure it out for a while now, and I still don't know the name of this game...
(I think it played outside of actual gameplay, like at the title screen, level select etc.), and I'm pretty sure the rest of the soundtrack is based on classic Christmas music as well.
Edited by Eggy0Anyhow, as you might have seen from the title, it's a Christmas-themed 2d platform game. It was for the PC, and I'm pretty sure it dates from either the 90s or the early 2000. The game stars a Christmas elf as the character the player controls, and it has 25 (or maybe just 24) levels. I think the first level(s) is/are a place made of candy.
The main mechanic of the game is that your player character possesses a candy cane that works like a jetpack (as in you can propel yourself into the air with it, complete with a combustion flare if I'm not mistaken). The jetpack candy cane also has some sort of gas meter in the top left corner of the screen which depletes as you propel yourself into the air, and instantly replenishes when you touch the ground. Collecting candy canes scattered across the level you're playing on increases the gas amount you have (allowing you to propel yourself more before you run out of gas and start falling), which resets upon exiting/completing the level.
I think that, in each level, you have to collect multiples of an item to spawn a door that would lead to the next level being unlocked, but I can't remember the exact item you have to collect. Perhaps something that pertains to Christmas like everything else in the game.
The one music I can remember playing in the game is a reiteration of Joy to the World
That's all I can remember right now. I hope someone will be able to figure it out for me as I couldn't and searching didn't help much.