Does diving after a take-off help gaining altitude?
When an aircraft can barely climb after take-off, it may be flying on what is called 'the backside' of the power curve. 

Flying at very low speed:


costs a lot of power because of high induced drag
means that the engines are inefficient because they are optimised for higher speeds.


As a result, the excess power available, that is the difference between the required power and the maximum available power, is very low when the aircraft flies close to the minimum flight speed. And the excess power is what is needed to climb the aircraft. 



When the aircraft is flying below the speed at which the best angle of climb is achieved (Vx), it will be able to improve its climb angle by accelerating. To do so, the aircraft needs to use available power to accelerate, instead of using the power to climb. 

Stopping the climb or even descending a bit will convert the available excess power and optionally height (potential energy) into speed. While this initially does not give extra clearance above the obstacle, the available excess power increases drastically. This in turn gives the ability to climb at a much steeper angle, allowing the aircraft to clear the obstacle.