Here's some thoughts, but I don't claim to be an expert...

What year is your car?

Are you getting any dashboard warnings (the big exclamation mark)?
Can't quite be sure of what you are saying... are you saying the
pump doesn't run at all, eg, you cannot hear the pump run momentarily
when you first turn on the ignition? (I'm assuming that is the symptom)
...Or are you saying the pump doesn't seem to be putting out a
sufficient flow to keep the engine operating?

My Haynes manual shows the fuel pump relay as a brown relay beneath the
passenger dash. A brown wire from the battery feeds it... check that
brown wire for 12v (wire to ground) at the relay with a multimeter.

If you have power there, maybe the ignition switch is not switching
the fuel pump relay on through the "ignition on" relay (blue) or the
main relay (purple, I think).

The main relay should send battery power to operate the fuel pump
relay via a  white w/ brown trailer wire. I'd be tempted just to
replace all three of those relays, but perhaps you can test them
(easy for me to say, I don't have to do it... there's a procedure
in the Haynes manual that (with the relay removed) consists
of checking continuity between the switched "power" terminals
while applying 12v through the control "energizing coil" terminals...
the procedure is a bit too wordy for me to retype here).

The wiring diagram shows a straight shot from the FP relay to the
fuel pump (blue wire w/ red tracer, then a black wire from the
pump to chassis ground). Check the black wire (disco'd at the pump)
resistance to a known chassis ground with a multimeter on the ohms scale
(should be zero resistance). At the fuel pump, check the blue/red wire
(while
the ignition is being switched on) for 12volts across to a known
good chassis ground during the first second after the ignition is
switched on.
If no power, check the same wire at the fuel pump relay end.

If the blue/red wire is not being powered, and you know the "ignition on",
main, and fuel pump relays are good, and that there is power to the blue/red
wire at the fuel pump relay, then it's a matter of determining where
the power feed is broken....

If you didn't already replace the FP relay, try swapping it with another
similar relay to eliminate it, ie, put a known working relay into the
spot where
the brown relay normally goes and check for power flowing out the blue/red
wire. If you have power coming into and out of the FP relay, but no power
at the pump, then you need to check for a bad harness connector on the
harness that carries the blue/red wire.

Beyond that, if the fuel pump is not being powered, it seems to me
that the ecu is seeing some problem and not allowing the FP relay
to be switched on, hence my question about dashboard warnings.