These are the only other relays I can find on the car apart from in the fuse box. These are mounted in a blackbox on the passenger side in front of the scuttle plate.
BTW Not sure if you remember my car, it's a Charcoal 205 MI16, which you did the rear beam and handbrake cables on a few years ago.
You said for a 405, not a 205. Very different. The fuel pump relay for a 205 will either be in a black box, behind the LH headlight (Jetronic) or up near the ECU, under the dash (Motronic). I can't recall any other oem locations/variations. The only other possibility could be if you're running a full 405 Mi16 loom. Regardless, look for a thick white wire, #76 or #76A.
I've had this problem with two Mi16s that had been sitting for a while. I hardwired 12v direct to the pump, which got them working. From then on the wire could be removed and the pump worked as normal.
What Graham is saying sounds like pump dried perhaps with some tiny residue left behind by drying fuel that locked it up a bit, just enough that it doesn't want to move that easily. Happens in carbies as well and you can have the float stuck or some such. Hot wiring it probably gives it more power, maybe just enough that it starts spinning again.