Hi Steve
The smoothing cap is the round silver can about 35mm in diameter fitted through the chassis, it is actually Two capacitors C19 smoothing and C21 Reservoir in a single housing It is unlikely to be faulty.
Do you have the service data for the set? (Available from the link at the top right of your forum page) it would be worth checking the resistors to see that none have drifted out of tolerance. you will need to desolder or snip one end out of circuit and also be aware that your fingers on the voltmeter probes will affect the readings.
I see from your photos that some pins (3 & 4) on V4 UL41 output valve socket have been used as anchor points for part of the circuit, this is how Bush wired it originally but is far from ideal and could well be the source of some of the hum you are experiencing. if you search threads on this forum you will find reference to a modification to the wiring to avoid this. I will search and see if I can find a drawing or photograph to illustrate this modification.
Don't expect too much from the radio, there are not many MW/LW stations operating now and those that remain are often drowned in a sea of background noise,
Please be aware as others have said that this is a
Live chassis make absolutely sure that the set is unplugged before you work on it.
Kind regards RT