There is no point in changing to
p6 on a normal Raspbian install. Those are partition numbers starting from one and there are only 2, so
p6 will never work. NOOBS uses additional partitions (and it is not impossible for the numbering scheme to have intentional gaps, but Raspbian does not -- see comments at bottom).
The original error is probably because the partition table was corrupted. If you have another linux machine or something you can run
fdisk on stick it in that and look at the output of
fdisk -l to see what happened. Unfortunately, unless you have a backup copy or the original image you installed from, figuring out how to set up/correct the partition table will be awkward because the boundaries have occasionally changed.
If you don't have something with
fdisk, there are probably other tools that could be tried on any operating system, but again, you will have to use them to investigate the state of the partition table first.
Even if you do restore the table, there is no guarantee the data on the second partition will be okay and you should make sure to run
e2fsck on it before you do anything else. However, the fact that the system still boots is a good sign -- the corruption might be very limited.