Noise when steering/accelerating after transmission replacement

Hello, golf experts!

I have a question for you all.

In my Golf IV (AJM TDI, 115 HP MJ2001), which has 63,000 km on it, the third (6-speed) transmission has now been installed.
The original one made a very loud noise, so I got a new one under warranty. However, the new one also made very loud noises in the 3rd, 5th, and 6th gears (a whining sound). The current (modified) version is working very well so far.

The current problem:

After each transmission change, I noticed the following:

As soon as I try to start moving in first gear and turn left, I hear this strange noise (it's almost like something is rubbing). However, I don't hear it when I turn right. After the transmission was replaced in the first gear, the noise disappeared after approximately 1000 km.

'Now, I'm hearing the same noise with my current gearbox.' It's been there for 14,000 km now, and it's not getting any better (I thought it would go away on its own).

Here's another one: This noise is only audible when the VEHICLE IS COLD, and as soon as I disengage the clutch while turning, the noise disappears; it reappears when I re-engage the clutch.

He said it was the power steering pump, and that it's normal, but I don't believe him.

It would be great if you could help me.

Thank you.
Ann Kathrin