Question

if a score is change, will the mean be affected?
if a score is added in the data set, will the median be changed?

Answer 1

changing a score will change the mean

adding a score will change the median
... unless the score is equal to the median

Answer 2

The more a score is changed, the greater the effect on the mean, especially with small samples.

Assuming the score is not at the median, the median will change slightly. The change might be greater for small samples.