The features of resonance phenomena are an important problem, both experimentally and theoretically. For the integrable systems, resonance in solitons may lead to various types of new excitations such as the breathers [
1], the soliton fissions, the soliton fusions [
2], the rational-exponential waves [
3] and so on. The soliton molecule, which can be treated as the soliton bound state, has attracted considerable attention. The soliton molecules were first predicted theoretically in the framework of the nonlinear Schrödinger-Ginzburg-Landau equation [
4] and the coupled nonlinear Schrödinger equations [
5]. Recently, a new velocity resonance mechanism is introduced to form soliton molecules [
6,
7]. For velocity resonance, high-order dispersive terms play a key role in the nonlinear integrable systems [
6]. Based on the velocity resonance, the soliton molecules of the (2+1)-dimensional fifth-order Korteweg–de Vries (KdV) equation [
8], the complex modified KdV equation [
9] and the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation [
10] are constructed by using the Darboux transformation and the variable separation method. Meanwhile, some novel interactions between soliton molecules and breather solutions, and between soliton molecules and dromions, are presented by the velocity resonance mechanism [
8–
11]. The interaction between solitons and other nonlinear excitations plays an important topic [
12–
20]. The consistent Riccati expansion (CRE) or the consistent tanh expansion (CTE) method can be applied to find these types of solutions [
21]. The method has been applied to various nonlinear systems, including the modified Kadomtsev-Petviashvili (KP) equation [
22], the modified KdV-Calogero-Bogoyavlenkskii-Schiff equation [
23], the supersymmetric integrable systems [
24] and the non-integrable cubic generalised KP equation [
25]. In this paper, the main purpose of our work is to construct soliton molecules, interaction between a soliton molecule and one-soliton, and interaction between solitons and cnoidal periodic wave solutions for an extended modified KdV (mKdV) equation.