The fuel pump, relay or anything of that sort isn't the issue.

20psi with vac, 43psi (3 bar) without.  That's the bit that's not right.

Hey Adam and Nyet thanks for replying. I agree with you Adam. Nyet, the Bentley states that at idle, fuel pressure should be no less than 36 psi with the vacuum hose attached to the fpr and 44 psi with the vacuum hose plugged and unattached to the fpr. and then leak down after 10 minutes after the engine turned off, fuel pressure should be no less than 29psi. The weird thing is that when I first start the car fuel pressure is to spec then slowly drops and ends up at 20 psi. I noticed it is failing the leak down also, but I didn't find any leaks anywhere. The Bentley suggests the fuel pump check valve is the cause of that but I can't find a check valve anywhere in the system so maybe its inside the pump? or inside the fuel filter? But that being said I had both problems before I changed the pump and fuel filter. Then Bentley also suggests that there may be a clog in the supply lines between the pump and the motor which is reducing the pressure getting to the fuel rail. I think my car got the COVID-19 or something. If I had a compressor I would just blow out the supply line but the ex-gf stole that also...long story..ok I have to add that she stole my tool chest and a number of my other belongings but left the tools that were inside...crazy right? Anyway the solution still eludes me..

Ok so it took me a bit to get situated but now I can post more info. So Nyet, I have answered your questions in order and then added some more info after.

Manifold vac @ idle = 19-18 in/Hg with AC off and 16-18 in/Hg with AC on.

Yes, the fuel pressure is @ 3bar without the vacuum line connected to the fpr.

There is only one return line, which goes directly from the fuel rail back to the tank. As far as I can tell it is fine. It is returning the fuel like it is supposed to.

 The fuel pump I installed is identical looking to stock OEM unit, it installs into the same basket assembly. Perfectly plug and play. But it is rated to 265LPH @ 13.5 volts and 3 bar.   Unfortunatly, I have yet to find the info on where exactly the check valve is in the fuel system for my car. I did try to blow through the old fuel filter backwards and I could not blow through it, so there may be a check valve in the fuel filter as well as the fuel pump.

added info - When I first start the car, the fuel pressure will drop to 36 psi while the vacuum line is connected to the fpr. Which, according to the Bentley repair manual, is exactly what is supposed to happen for my car which is a 2002 mk4 golf GTI with the 1.8T.  Now, what happens over a time span of about 10 -15 minutes sitting at idle is the fuel pressure gradually drops to 20 psi. Its very slow to do this, like not sudden at all.

So I purchased a nice multi-meter which will check DC amp draw. What I found was that when I first start the car the fuel pump draws between 6.5 and 6.6 amps. But as time goes on and fuel pressure drops down to 20 psi as I already explained, the amp draw drops also - to about 6.3 amps. I checked amp draw at the fuse block on top of the battery and had an amp draw of between 8 and 9 amps @ the lead for the fuel pump. This is all while the battery voltage is atleast 13.5- 14.05 volts.

Being that the pump is supposed to draw approximately 8.2 amps @ 36 psi per its spec sheet, at first led me to believe that the pump is weak. But I had this problem of low fuel pressure before and is why I replaced the pump in the first place!

So now I am starting to believe I have a wiring issue, being that I already replaced the fuel pump relay(409). The wiring for the fuel pump sits unmolested at this time. There is only 2 things that I feel I can do which would be to install the aftermarket relay I have for the fuel pump, which will tell me for sure if the stock wiring for the fuel pump is going bad. I know that copper begins to deteriorate after about 10 to 15 years so maybe there is a good amount of resistance build up somewhere between the fuel pump connector and the fuse on top of the battery for the fuel system. And #2 change out the feed line from the pump to the fuel rail with the -6an line I have sitting here which will cost me more money for fittings and time to modify the fuel pump assembly which I want to do anyway for when the other motor is finished.

My thoughts: IF the feed line is clogged why is the pump drawing less amps than it is rated for? If the return line is clogged why does the pressure drop over time instead of build up to high? Either scenario, I feel like it would draw more amps because it's having to force fuel where it doesn't want to go. Also if I clamp the fuel line right before the fuel rail and have a helper open the door to the car, the fuel pressure gauge shows slightly over 100 psi so I don't think the pump is the issue at all, at least its ability to make pressure. ..and the pump turns off after 2 seconds when you open the door. So it builds pressure rather quickly in my opinion.
I think I will see what happens when I disconnect the vacuum line to the fpr, plug it so no vacuum leak and let the car idle for 10- 15 minutes and see if the fuel pressure drops away as it has been. I believe this will tell me that either the fuel pump gets weak or the wiring is going bad. I will monitor amp draw during this test also.

Thoughts? Questions? Comments?

voltage at the pump is 13.5-14.05V depending on if the cooling fans are running or not.

I thought the same thing.. this issue has me flustered.

I haven't performed my next test yet that I explained above but I will tomorrow err.. later today rather.

I don't know what else to do other than install the relay with new wiring using the oem + lead for the pump as the signal for the relay so it works like oem and run power directly from the battery to the relay heading to the pump. Hell while I'm at it maybe I should get that IE fuel rail I've always wanted. not that I expect it to fix the problem..but whatever..it will damn sure look cool.