A faulty capacitor can not cause transformer hum.
What fault of the capacitor?
Generally, electrolytic capacitors lose their capacitance and it can not cause the hum of the transformer.
If short circuited, the capacitor will blow the fuse.
High leakage current is also impossible because the capacitor heat up tremendously and explode in a short time.
By cons, this can be caused by a faulty diode bridge with an open diode.
Try to check if the transformer continues to produce hum with secondary connections disconnected.
Does the transformer works abnormally hot without load?
Have you tried to tighten the bolts of the laminations?
To impregnate the transformer again, use a special varnish for this use.