Can u amend an application (i.e. send in a second more complete cover letter) or is it not advised ?

I wrote a cover letter for LSE and think it actually wasn't that great, and would love to send in a new format that i've been sending out to other schools which is more complete and just all around better

Yes i think you can by submitting the new info via the link provided on the e mail acknowledging the original application. Im on the waiting list so i sent on an updated personal statement which admissions said they sent on to the dept. Good luck!