Using $448.0 \times 10^6$~$ψ(3686)$ events collected with the BESIII detector, an amplitude analysis is performed for $ψ(3686)\toγχ_{c1}$, $χ_{c1}\toηπ^+π^-$ decays. The most dominant two-body structure observed is $a_0(980)^{±}π^{∓}$; $a_0(980)^{±}\toηπ^{±}$. The $a_0(980)$ line shape is modeled using a dispersion relation, and a significant non-zero $a_0(980)$ coupling to the $η^{\prime}π$ channel is measured. We observe $χ_{c1}\to a_2(1700)π$ production for the first time, with a significance larger than 17$σ$. The production of mesons with exotic quantum numbers, $J^{PC}=1^{-+}$, is investigated, and upper limits for the branching fractions $χ_{c1}\to π_1(1400)^{±}π^{∓}$, $χ_{c1}\to π_1(1600)^{±}π^{∓}$, and $χ_{c1}\toπ_1(2015)^{±}π^{∓}$, with subsequent $π_1(X)^{±} \to ηπ^{±}$ decay, are determined.