Is this a weed?

Slinky

This is a well established alkanet we attempted to dig out today. Some parts of the roots are over 2 inches thick and they go down a long way. We plan on planting a new fruit tree in the space, but have decided to put it in a pot for a while to see what starts coming up again from bits of alkanet we've missed.

Dustyevsky

Painting the newly-cut tops of anything with a strong glyphosate mix usually ensures no re-growth. That even applies with plants as large as a 30cm diameter cherry tree. Cover the cut surface with plastic to keep out the weather, and with large woody plants, ream out the clean cut surface with the chainsaw first.

Zanderman

Dustyevsky said:

It's immaterial whether the plant is a foxglove or alkanet. It's taken up residence at the base of a recently painted wooden structure, so if that has value, the plants should go. 

While they stay, the leaves will create shade, reduce air movement, keep the wood damp, and harbour creatures that exploit any rot.

Something tells me they've probably already gone!

No need for the plants to go just because they're next to a painted fence. If the plants are wanted - and the OP's question asking if the plants were weeds implies they're not necessarily not wanted, then why not stay. Alkanet, in flower, would be very attractive against that sort of background. The fact that the vegetation might shorten the life of the fence (but only a little, to be frank, and hardly at all if the fence paint is regularly re-painted) is secondary to the question what the plant actually is. It's the fence, surely, that is immaterial here. And, as Slinky says above, removing alkanet might be virtually impossible, unless the fence is removed first!.

Dustyevsky

Zanderman said:

Dustyevsky said:

It's immaterial whether the plant is a foxglove or alkanet. It's taken up residence at the base of a recently painted wooden structure, so if that has value, the plants should go. 

While they stay, the leaves will create shade, reduce air movement, keep the wood damp, and harbour creatures that exploit any rot.

Something tells me they've probably already gone!

No need for the plants to go just because they're next to a painted fence.

I agree, there's no need for the plants to go if the OP thinks they look attractive. There's often a trade-off between attractiveness and utility. 

OP described the positioning as 'awkward' and I took a cue from that. Whether the plants are alkanet or foxgloves, they will eventually flop over the pathway.