How to trigger Britek PS-300H Studio Flash with Nikon D40 via sync cord?

I'm using a Nikon D40 camera (with AS10 type flash adapter) to trigger my Britek PS-300 Studio Flash via a sync cord. When it is on manual the flash is not triggered. However, when the camera is placed on automatic the flash goes off every time. I try to shoot the cameras with the "pop up" down but after shooting a few photos it pops up (by itself) although the flash adapter and sync cord is connected.
How do I get the Britek PS-300H studio flash to fire every time on manual. I use camera settings of: 640 f5.6 and iso Auto. I tried several combinations of camera settings but none worked consistently well.

What's the maximum flash sync speed of your camera? If you don't know and can't find it in the manual, try 1/125s. It should fire even in manual mode.

With the D40 the x-sync speed is 1/500th second. With the camera in M, normally the built -in flash would never pop-up by itself. It would in Auto.
Even with a flash in Manual mode, when it's turned on the camera should detect it. With the camera in M, it should limit the shutter to 1/500th second even if it had been turned up to a faster speed with a flash absent. In Auto the camera should choose 1/60th second by default, and it should abstain from popping up the flash. It works that way with a Nikon flash on the hotshoe. Somehow the camera is treating your flash as if it is absent or turned off.