In this paper, by the method of comparing coefficients and the inverse technique, we establish the corresponding variate forms of two identities of Andrews and Yee for mock theta functions, as well as a few allied but unusual $q$-series identities. Among includes a new Bailey pair from which a product formula of two ${}_2φ_1$ series is derived. Further, we focus on two finite $q$-series summations arising from Andrews and Yee's mock theta function identities and expound some recurrence relations and transformation formulas behind them.