I understand how to use square brackets in quotes when adding full words, but if I am quoting something that was misspelled initially and want to correct the spelling, is it proper to do this? For example, I am quoting someone who said ”Additional, _____”, would it be appropriate for me to type ”Additional[ly], _____”?
It seems you have two options, if that part of the quote can't be omitted:
Do what you suggested -- Additional[ly]
Add [sic] after the word -- Additional [sic]
The first is probably the clearest way to communicate that you're quoting verbatim, yet also know there was an error to correct. As for the latter, be aware that many people associate a certain snarkiness with [sic], which is less pronounced by just correcting the error in brackets.
One way of showing that you, the person quoting text with errors in it, has noticed that they are errors is to use the Latin word "sic" meaning "thus", or in this context "don't blame me, that is what the ignoramus who I am quoting wrote".
So, in your example you would write:"Additional [sic],..."
This case seems clear, but, as a warning, if you are going to sic something, do check that it really is an error. I have certainly seen sic applied to things that were quite correct. Who's the ignoramus then?