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ЧЁРСТВЫЕ ИМЕНИНЫ coll NP fixed WD) the day following the celebration of a person's name day, when people may come over to offer belated congratulations, eat food left over from the party etc

the day after s.o. 's name day

приходить на чёрствые именины - come (go) over s.o. fc place for leftover birthday cake

come over etc to wish s.o. a belated happy name day.

The idiom refers to the fact that on the day after a party the cakes are already stale («чёрствые»).