Hi Christine
I have 
Mary Pate bpt 14 April 1765, Lesmahagow - parents Thomas Pate and Isobel Meikle
No James bpt Shotts but
a James bpt. 14 may 1762,  Lesmahagow - parents Thomas Pate and Jane Telfer
 a Janet,  bpt. 23 April 1758 Lesmahagow - parents Thomas Pate and Jane Telfer
The only child of a William Peat showing is:
George Peat, 13 March 1769, Shotts  parents William Peat and Jean Rankine.
I searched on P*at* and Lesmahagow and Shotts 1741 to 1770
I'll go see if I can find anything else  

Gadget