I am a poor mathematician looking into the Sars-Cov-2 genome. Is there a proper definition of the word homoplasy? Is it just mutations that happened independently or does there have to be indications that the mutations give an improvement in fitness?
Homoplasy just refers to the fact that the mutation happened independently. While several sources only discuss homoplasy as a phenotypic characteristic, and in contrast to homology (and thus suggest an adaptive implication), in general, there is no assumption that the mutation gave increase in fitness (see this paper).
According to Wikipedia (obviously not the final authority on scientific questions, but often reliable), homoplasy occurs
when a trait has been gained or lost independently in separate lineages over the course of evolution
There's nothing in this definition about the fitness implications of the trait (beneficial, deleterious or neutral).
"Trait" is a more problematic/nebulously defined term, because you have to carefully define what the trait means in different lineages, but in this case it's probably relative to aligned sequences, e.g. "a G at position 575 of the aligned sequence", or "a deletion from positions 575-580".