Using Cumberland Highway IAMS website confirms no footpath here. 
Signs are marked upon entrance to the path stating there is no footpath here. 

Discussions on Facebook confirm local concerns over legitimacy of the footpath being closed dated back to 2021 (ref: https://www.facebook.com/spottedcarlislenew/posts/the-footpath-cycle-path-leading-from-burgh-road-industrial-towards-the-river-ede/1829907340499487/). 
Archived link for Facebook post (ref: https://archive.is/wLYfU).

A document concerning proposed modifications and inclusion of the footpath can be found here (ref: https://www.carlisle.gov.uk/Portals/0/Documents/PDF/PF10971%20Burgh%20Road.pdf?ver=hh9E90KPF_jcGcHr8Oh8Bg%3d%3d) however no correspondence can be found and seemingly no updates made to the public right of way/footpath map found here (ref: https://cumberland.highway-iams.uk/Live/PBLC/PIP/?cg=networks) 