Full answer from OP
The credit for right answer has been given to @malioboro, but @SwiftPanda deserves a big shoutout as they were the one that cracked the majority of the clues.
As the solution is scattered across a few different answers and comments, I'm leaving the complete solution here as a single answer.
Where to start?
The first clue is the rhyming scheme being broken (lake/cake, cold/79), so we are looking for something which rhymes with 'cold' and is associated with the number 79. A quick look on the periodic table of elements shows us that gold has atomic number 79 and so the periodic table is probably key to solving at least part of the puzzle
My first and my last is one third of a lake
There aren't any lakes on the periodic table, but lakes are usually full of water, which is H20. One third of a water molecule is oxygen, so our first and last letter is 'O'
My fourth is the start of what can be enjoyed with a cake
The chemical symbol for titanium is 'Ti', which sounds like "tea". The start of the word "tea" is 'T', and so that must be our fourth letter
My third is the end of the sound made when cold
This time we need the chemical symbol for bromine, which is 'Br' and sounds a lot like "brrrrrrr" (the sound one would make when cold). The end of 'Br' is our third letter, 'R'
And second is second of the second where one can find 79
We've already worked out that "where one can find 79" is the periodic table, so the second element is helium, which has the symbol 'He'. The second letter of that is 'E', which must be our second letter
Once you've found all five (and don't get too heated)
In order, our letters are: O E R T O (the "don't get too heated" was just to make the rhyme match with the final line)
Look in the gaps to find how you'll be treated
By viewing our 5 letters like this: O_ER_T_O_, we could fill in the blank spaces using the letters 'P', 'A', 'I' and 'N', to make the word OPERATION.
So what is O_ER_T_O? - it's the word "OPERATION" with the letters that spell "PAIN" removed from it, which makes it a...
(I still need to find a new GP though...)