The Richtmyer–Meshkov (RM) instability is numerically investigated on an unperturbed interface subjected to a diffracted convergent shock created by diffracting an initially cylindrical shock over a rigid cylinder. Four gas interfaces are considered with Atwood number ranging from $-0.18$ to 0.67. Results indicate that the diffracted convergent shock increases its strength gradually and reduces its amplitude quickly when it propagates towards the convergence centre. After the strike of the diffracted convergent shock, the initially unperturbed interface deforms with a bulge structure at the centre and two interface steps at both sides, which can be ascribed to the non-uniformity of the pressure distribution behind the diffracted convergent shock. With the decrease of Atwood number, the bulge structure becomes more pronounced. Quantitatively, the interface amplitude experiences a fast but short growing stage and then enters a linear stage. A good collapse of the dimensionless amplitude is found for all cases, which indicates a weak dependence of the growth rate on Atwood number in the deformed shock-induced RM instability. Then the impulsive theory is modified by eliminating the Atwood number and considering the geometry convergence, which well predicts the amplitude growth for the deformed shock-induced RM instability. Finally, the underlying mechanism is decoupled into three parts, and it is found that both the impulsive pressure perturbation and the geometry convergence promote the growth of interface perturbation while the continuous pressure perturbation inhibits the growth. As the Atwood number decreases, the impulsive perturbation plays an increasingly important role, which suggests that the impulsive perturbation dominates the deformed shock-induced RM instability at the linear stage.