The relative hyperbolicity of one-relator relative presentations

We prove that if $G$ is a free-torsion group and $w(t)$ is a word in the alphabet $G\bigsqcup \{t^{\pm 1}\}$ with exponent sum one, then the group $<G,t\mathrm{\mid }(w(t){)}^k=1>$, where $k\ge 2$, is relatively hyperbolic with respect to $G$.