firing a cannon

By relative to the barrel, it means from the point of view of the barrel, if it were stationary then the projectile has a speed of 306m/s. Now, it says in the question that the gun "can move horizontally against a recoil mechanism", i.e. it is not necessarily stationary. Therefore, if say the gun were moving say 6m/s west and and the projectile 300m/s east, the relative speed is 306 m/s.

Now, in terms of impulse, we only need to concern ourselves with the horizontal impulse, not the vertical impulse. This is because all vertical impulse is imparted from the cannon to the ground, so overall none is transferred to the cannon. However, because the cannon can move horizontally then impulse can be transferred to it in the horizontal direction.

The horizontal relative speed of the projectile, as stated is 306cos60 = 153. You have from newton's laws that total momentum before (which is 0 as all objects were stationary) is equal to the total momentum after. Therefore, we have $p_{cannon} + p_{ball} = 0$, where p_object is the momentum of that object. Expand out p as p = mv, and then note speed of cannon + speed of ball = relative speed. Then, substituting this back into the original equation, you can solve for the speed of the cannon, and thus the impulse imparted on the cannon