I had the same problem on a Singer 327P, also caused by a shorted capacitor connected across the porcelain device. The porcelain device probably is a rheostat, but I can't figure out how it works internally so I'm not sure it's as simple as that.
My capacitor was 0.1 uF (marked HUNTS / Cap 0.1 uF / 250v A.C WKG. / M.P / WA 71 IN1) and it was connected straight across the porcelain device. You should replace it, otherwise the porcelain device could fail very quickly due to sparking, as mentioned by others, and it will generate lots of interference.
You need to use a capacitor that's rated for continuous AC mains voltage - either 250V AC or 125V AC depending on your country. These should be available from DIY electrical parts shops.
The original question had two 0.04 uF capacitors. Depending on how they're connected, they could just be noise suppression capacitors and removing them completely might not affect reliability, but I recommend replacing them as well. Same rule (continuous AC mains rated) applies.
In reply to the question about how the pedal is wired, mine has just two wires from the machine to the pedal, and they connect to the terminals of the porcelain device, with the capacitor also connected directly across the terminals. If your pedal has a third wire it might be a safety earth, to be connected to the pedal's exposed metal parts (case etc). Check for electrical continuity between the sewing machine plug earth pin and the wires at the pedal to find which one is earth. If there are two suppression capacitors they may connect from the two sides of the porcelain device to earth (one capacitor on each side). Also the third wire could be Neutral not Earth, in which case it would connect to the suppression capacitors but it shouldn't be connected to any exposed metalwork.
In response to the question about blowing the mains fuse when the pedal is left up for a few minutes, this could be a faulty suppression capacitor (but not if the only suppression capacitor is connected straight across the porcelain device), but is more likely to be the porcelain device itself. When you leave the foot pedal fully up, depending on the mechanical adjustments inside the pedal, there may still be some current flow through the porcelain device, but not enough to turn the motor. In this state the porcelain device will heat up, and parts may expand and/or go soft, so things that shouldn't touch together may touch together, blowing the fuse. You need to open the pedal up and leave it plugged in with the lights off, and try to see where the spark comes from when it fails. You could also try a low-current resettable cutout in line with the sewing machine so you don't have to replace your fuse for each test run, or to avoid tripping any breakers or fuses, connect a light bulb (100W or more) in series with the sewing machine, but you have to be careful with mains voltages, and preferably have an experienced friend do this.
Anyone reading this thread, please feel free to email me (the address is with my name). I'm an electrical/electronic engineer but I don't have much experience with sewing machines, but I'm keen to learn and help if I can.
Kris Heidenstrom - kris at heidenstrom.gen.nz