Based on the values given in the question, I make $R_p=0.1195$ before the 1.6% increase and $0.1215$ after the increase. So $R_2/R_3=22/1.2=18.33$ whereas $R_1/R_p=2.4/1.1195=20.08$ or $2.4/0.1215=19.76$.
The bridge is not exactly balanced in either case. Consequently there will be a non-zero PD between points A and B. The value of this PD is affected by the internal resistance of the cell, which will reduce its terminal PD, and therefore the PD across AB.
So I think the answer is that the PD between A and B, and therefore the reading on the voltmeter, does change when the internal resistance of the cell increases. I do not think "approximately balanced" is good enough to avoid confusion in an examination.
Futhermore, the question says "Explain without calculation..." I do not see how it is possible from the previous calculations to conclude, "without (further) calculation", that the bridge is "approximately balanced" - which is the only case in which a change in the terminal PD of the cell would not affect the voltmeter reading.
The increase in the resistance of the probe in Q 4.5 is also very confusing : are we to assume this increase applies in the bridge circuit, or that it is a hypothetical change?
It appears to me that the question is faulty and should not have gotten onto the examination paper. The fact which it asks you to explain (that the reading on the voltmeter "does not change") is not true and it is not obvious how anyone could know whether or not it is true "without further calculation". For an examination board with such a high reputation, this is very poor.
AQA has admitted faults previously, eg in June 2015, but I cannot find any acknowledgement of error in this case.
The mark scheme acknowledges that the bridge is not exactly balanced, and suggests that since the PD across AB is "very small" the voltmeter may not be sensitive enough to to detect the change. A very sensitive voltmeter or galvanometer is usually used in Wheatstone bridge measurements. The question gives no hint that the voltmeter might lack sensitivity, such as by asking why the voltmeter reading "might not" change, or that the change "might not be detected, instead of "does not change", which implies it will not change for any voltmeter.