The only information I have found on the 13th Battalion, Lincolnshire Regiment is a list of 13 names in the CWGC records.
BRODDLE, O. Private, 36868. 13th Battalion, Lincolnshire Regiment. Etaples Military Cemetery, Pas-de-Calais, France. Sunday 22-04-1917. Age 30. Son of George & Emma Broddle. Husband of M. Broddle of Roxburgh Street, Hull, Yorkshire, UK.Native of Anlaby Common, Hull.
HAIGH, A. Private, 34341. 13th Battalion, Lincolnshire Regiment, Transferred to :- Private, 211292, 409th Agricultural Company, Labour Corps. Newport Cemetery, Lincoln, UK. Saturday 09-11-1918.
MAYS, S. Private, 36959. 13th Battalion, Lincolnshire Regiment, Transferred to :- Private, 30875. 52nd Company, Labour Corps. Locre Hospice Cemetery, West-Vlaanderen, Belgium. Saturday 20-10-1917. Age 24. Son of M.J. Mays of Adelaide Street, Hull, Yorkshire, UK. Husband of Vanetta Mays of Carlton Terrace, South Boulevard, Hull. Stone Inscription :- "Safe In God's Keeping Just Out Of Sight".