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Please don't do anything to bring shame on the Matsuno family name.
Literal
Matsuno [genitive-の] family-name [object-を] stain [thing-こと] [subject-が] not-exist [as-ように] please.
A formal request from a parent or elder asking a family member to protect the family reputation. けがす ('to stain, defile, bring disgrace to') is a transitive verb often paired with 名 ('name') or 名誉 ('honor'). The frame ~ことがないようにお願いします ('please [make sure] that X does not happen') uses the combination ~ないように (purpose/goal: 'so that X doesn't') + ことがない (nominalized 'the event of X not happening'). Together: 'please [ensure] that the family-name-staining doesn't happen.' A dignified and indirect request.