A group obtained from a nontrivial group by adding one generator and one relator which is a proper power of a word in which the exponent-sum of the additional generator is one contains the free square of the initial group and almost always (with one obvious exception) contains a non-abelian free subgroup. If the initial group is involution-free or the relator is at least third power, then the obtained group is SQ-universal and relatively hyperbolic with respect to the initial group.Henceforth, the symbol |u| denotes the number of letters t ±1 in the word u.Relatively hyperbolic groups have many good properties. For example, they are SQ-universal (apart from some obvious exceptions) [AMO07], the word [Far98] and conjugacy [Bum04] problems are solvable in such groups (under some natural restrictions). The same is true for many other algorithmic problems. More details about relatively hyperbolic groups can be found in book [Os06].It turns out that the torsion-freeness condition in Le Thi Giang's theorem can be replaced by the absence of only order-two elements. Presently, the following theorem is the unique result about unimodular relative presentations in which torsion-freeness condition is weakened to the absence of small-order elements.