Can loving-kindness meditation help with self-compassion or a difficult relationship?

It can, and the two work differently. For self-compassion the evidence is reasonable: a 2014 review of 22 randomised trials of kindness-based meditation reported a small to moderate effect on self-compassion against passive controls, though the authors rated the quality of the reports as low to moderate. For a difficult relationship, treat the practice as optional and last. Goodwill toward someone who harmed you is not forgiveness, not reconciliation and not approval, and it is never a test of your practice. If bringing the person to mind is very upsetting, set them aside for now.