Alas LH's services have really been clawed back at LCY since BA turned up. MUC, HAM and worst of all THF have been eliminated entirely.

The LCY-FRA flights are part-operated by Eurowings, part-operated by Augsburg, part-operated by CityLine - seems to change and get mixed round regularly. You should get miles regardless of the carrier (avoided EW so far). The flight numbers are the standard LH4XXX and LH5XXX used for UK-Germany flights. Technically Augsburg flights shouldn't credit but they do (an agent at LCY, when I was telling her I hadn't got miles for the then Augsburg Airways LCY-FMO service, said it was something to do with the system seeing the FRA services as being operated "on behalf of LH". The flight number is apparently the giveaway for this.)

Cirrus Airlines operate LH's only other route out of LCY at the moment, to Leipzig. That has an LH9XXX flight number and AFAIK isn't therefore mileage earning except for Miles and More.