If the burner starts then I think the issue is with the FBH itself. My recent experience was different and I have a TD6 but photo below shows that the pump and burner etc are controlled via the FBH 'module' under the cover to the right of the connector block that feeds to the pump and injector etc. In my case the power transistor that operates the burner had failed but if it was me I would start by removing the cover plate over the connectors (removed in the photo) to check for water ingress/corrosion and whether you are getting a voltage for the pump - Its clipped at the front and rear (12/6 o'clock in the photo) with hooks so a suitable Right angle pick is needed to get down and under the edge of the cover clips and lift upward and outwards.

Click image to enlarge

@johnty - You're right there is a coolant pump underneath the valve assembly - this is the one that GAP is reporting as error 0x0304. I don't think it is faulty though - If I power it directly it runs fine. However, the ATC module is definitely not sending power to it, no matter what I do.

@Steve k - I'm inclined to agree with you Steve, my unit does start up but it never burns which sounds more like your experience. I think my next step will need to be a strip down of the FBH itself as you suggest.

Onwards and upwards !! Previous life - 2012 L320 SDV6 - Got divorced, say no more!
Previous life - 2014 L494 SDV6 - Got nicked, without keys, watched it go !
The Duchess - 2007 L322 TDV8 3.6 Autobiography with 270K miles on the clock
The Bonkers One - Audi SQ7 V8 4L TT - It will pull your face off.