Installed second home partition disc.

If it's truly a second partition and not an additional disk in an LVM setup, then no, data won't just flow into it...

If you've configured WHM to use this partition for new accounts, they'll be created there, but that won't have any affect on existing accounts on other partitions.

So, if the original is full, you'll want to move some accounts off it onto the new partition to give the remaining accounts some breathing room.