> if mass is assumed to be **constant,** the *velocity* of the centre of mass
> of the system has to be different after the collision for the kinetic
> energy to be different.
> 
> However, if the momentum of the system is conserved, *the velocity* of
> the centre of mass of the system **should remain the same**.

1) mass is **not** constant and velocity is different: in a completely inelastic collision the two objects (A: m =1, B  m = 2) stick together and mass becomes A+B = 3


![enter image description here][1]

Suppose that $v_a = 6m/s$ and $v_b = 0  \rightarrow E_k = 0.5 * 6^2 = 18, p_a = 1 * 6 = 6$ 
 
After collision velocity would be anyway **lower** as KE should be distributed among more mass, but some KE is lost in the crash. How much?

Momentum is conserved: $ p_{ab} = 6$ , from this datum you can calculate its velocity:
$$v_{ab} = \frac{6}{3} = 2$$ and $E_k = 0.5 * 2^2 *3 = 6$, two thirds of energy have been lost

> *how can there be a change in kinetic energy of the system if there is no change in momentum?*

A change of KE without a change of momentum is not only possible but very frequent, because as you noted p varies linearly and KE quadratically. You get the same product by a wide range of factors: 6 = 6*1 = 3*2 = 2*3 = 1*6 = 0.5*12 etc.

All these are same values for m*v, but as the figure for v must be squared, you get all different values between energy and momentum: KE =3, =6, =9, =18, =72 etc

I hope this clarified all your doubts

 [1]: https://i.sstatic.net/oYclE.png.