That’s the one piece of case law that I would have suggested.
The crux is the “place of work”. If the employer was to say to her, “Come into work”, where would the clmt go? Nothing suggests that “home” is the “place of work” – at least so far. Based on the info given so far I would be tending towards deciding that the mileage from home to client-site is incurred for the purposes of employment (wholly and exclusively). As such, it isn’t earnings.
NB: If there is any mileage from home to the employer, I would see that as being commuting and NOT allowable.
{Edited to add} CJSA/4721/2001 is another case where travel expenses were looked at. Bear in mind that the facts may be different and I haven’t checked to see if the JSA legislation is the same as for HB/CTB.