If I send in my laptop for repair, will I send the charging cable away?
Do not know if it's important but for safety's sake I would send it
It depends on workshop and also on mistakes. Some workshop wants to get the notebook complete with accessories, others do not. Ask better.
There serious workshop has all current power cord and work with laboratory power supply, so if the error is clearly not on power supply, it is not needed for repair.
If the notebook does not charge or does not do anything, then send the power adapter.