This drug may have the effect of speeding up your heart rate.
I guess it means that the drug results in speeding up heart rate.
The medicine had the effect of making me sleepy.
Similarly, the medicine results in user sleepy.
Both of them use a pattern like
A has the effect of X, where A represents the reason/cause and X represents the result.
However, this one seems more complicated.
The effect of the redundancies on morale has been extremely damaging
the pattern becomes
A has the effect of X on Y and A is omitted, is my understanding correct?