in many cases you could simply move the directory that is taking too much space to different directory then either make softlink or if that didnt work you can use mount --bind
for example if directory /var/cach/mygame is too big, move my game to /mnt/part2/mygames
then either do ln -s /mnt/part2/mygames /var/cach/ or mount --bind /mnt/part2/mygames /var/cach/mygames
the miunt option is not permanent so if it works, u will need to add it to /etc/fstab to make it permenent
depends on what you mean by “broken”. If broken means has bad sector or other hardware issue, then yes OP should transfer data to healthy partition and work from there. though it certainly won’t hurt if he attempted to recover data from broken partition (worst the HD dies and OP restore the backup on healthy HD) However he said “i broke my partition” which make me think its software issue, not hardware. in which case, would be faster to recover data directly after taking backup
this clones one parition to another which is fine if you have free partition with enough space lying around. My code was for taking compressed backup of the partition to be restored later if needed. Its less convenient but doesnt require as much space nor does it require an entire partition