lst = {{{4}}, {{7, 7}, {7, 7}}, {{5, 5}, {6, 6}}, {{2, 1}, {1, 1}}};
Pick[#, MemberQ[{___, 1}] /@ Map[Map[Last]@*Tally, #, {2}]] & @ lst
{{{4}}, {{2, 1}, {1, 1}}}
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