MySQL: How to fetch nth level child menus of a parent menu
A long, long time ago (Oct 24, 2011), in a galaxy far away, someone boldly asked

Find highest level of a hierarchical field: with vs without CTEs 

In my answer to that post, I wrote three stored procedures to find specific relationships


GetParentIDByID
GetAncestry
GetFamilyTree


If you use the code in GetFamilyTree, just pick the nth member of the output.

I have referred others to that post


Jan 31, 2014 : Recursive Query in MySQL using stored proceedure and CURSOR
Jul 11, 2013 : Recursive self joins
Dec 10, 2012 : MySQL: Tree-Hierarchical query


Give it a Try !!!