Look at the man page for apt-get.
Fix; attempt to correct a system with broken dependencies in place.
This option, when used with install/remove, can omit any packages
to permit APT to deduce a likely solution. If packages are
specified, these have to completely correct the problem. The option
is sometimes necessary when running APT for the first time; APT
itself does not allow broken package dependencies to exist on a
system. It is possible that a system's dependency structure can be
so corrupt as to require manual intervention (which usually means
using dselect(1) or dpkg --remove to eliminate some of the
offending packages). Use of this option together with -m may
produce an error in some situations. Configuration Item:
If I'm not mistaking mintupdate is just another front-end for apt-get and packages are saved in /var/cache/apt/archive. So if the apt-get -f doesn't resolve your problem you might want to delete the files in that directory (not the partial directory itself but if there are files in there, delete those too), run apt-get update and upgrade again. See where that leads you.