Suppose a metallic resistor is connected across a battery. If the...

The drift velocity of free electrons in a metallic conductor is given by $v_d = \dfrac{eE}{m}\tau$, where $\tau$ is the relaxation time (the average time between successive collisions).

If the number of collisions of free electrons with the lattice is reduced, the relaxation time $\tau$ increases.

As a result, the drift velocity $v_d$ of the electrons increases.

The electric current flowing through the resistor is given by $I = neAv_d$.

Since the drift velocity $v_d$ increases, the current $I$ will increase.

Alternatively, the resistance of the conductor is $R = \dfrac{ml}{ne^2\tau A}$. An increase in $\tau$ leads to a decrease in resistance $R$. For a constant potential difference $V$ provided by the battery, the current $I = \dfrac{V}{R}$ will increase.