We study effects of competing pairwise higher-order interactions (HOI) with alternating signs and exponentially decreasing intensity on critical behavior of the XY model. It is found that critical properties of such a generalized model can be very different from the standard XY model and can strongly depend on whether the number of HOI terms is odd or even. Inclusion of any odd number of HOI terms results in two consecutive phase transitions to distinct ferromagnetic quasi-long-range order phases. Even number of HOI terms leads to two phase transitions only if the decay of the HOI intensities is relatively slow. Then the high-temperature transition to the ferromagnetic phase is followed by another transition to a peculiar competition-induced canted ferromagnetic phase. In the limit of an infinite number of HOI terms only one phase transition is confirmed, and under the conditions of fierce competition between the even and odd terms the transition temperature can be suppressed practically to zero.