Deputy Defence Minister Hanna Maliar said the Ukrainian military had reclaimed about 20 square kilometers from the Russian forces around Bakhmut.
Neither Ukraine nor Russian forces have been able to take full control of the city of Bakhmut.
The identity of the grandfather figure Prigozhin referred to was unclear, but sufficiently ambiguous to invite speculation.
Wagner leader Prigozhin has been threatening to pull his mercenaries out of Bakhmut due to insufficient Russian support.
Deputy defense minister Hanna Maliar made her comments after Wagner Group leader Yevgeny Prigozhin said his mercenary forces would pull out of Bakhmut.