I didn't find any pension application for this man. His notes in Booth's Record of Louisiana Confederate Soldiers states...

Besson, T. D., Pvt. 26th La. Infty. Federal Rolls of Prisoners of War Captured Labadieville,
La., Oct. 27th, 1862. Paroled at Thibodaux, La., Nov. 4th, 1862.

Labadieville would be an unusual place for him to be and get captured at because at this time the 26th Louisiana Infantry was near Vicksburg, Mississippi.

Some possible reasons for his being home at that time could be he was sick or wounded and was home on furlough on on furlough for some other reason. I doubt he had deserted or his record in Booth would likely, though not definately, have listed that information.

I found the following information on Rootsweb. Is this your Great-uncle?...

Source: http://wc.rootsweb.com/cgi-bin/igm.cgi?op=GET&db=gbonnet&id=I32808

David Telesphor Besson
Birth: 25 JAN 1829 in Thibodaux, Lafourche Parish, Louisiana

Father: David Besson b: 1797 in Leith, Scotland
Mother: Marie Rose Arthemise Bourg b: 1806 in Plattenville, Assumption Parish, Louisiana

Marriage 1 Mathilde Bourgeois b: 1830 in Lafourche Parish, Louisiana

Married: 9 FEB 1854 in Thibodaux, Lafourche Parish, Louisiana

Children
Telesphor David Besson b: 21 JUL 1855 in Thibodaux, Lafourche Parish, Louisiana
Marie Georgine Besson b: 28 JUL 1858 in Thibodaux, Lafourche Parish, Louisiana
Josephine Anais Besson b: 18 FEB 1860 in Thibodaux, Lafourche Parish, Louisiana
Julia Besson b: 11 SEP 1861 in Thibodaux, Lafourche Parish, Louisiana
Amos Joseph Besson b: 7 MAR 1866 in Thibodaux, Lafourche Parish, Louisiana
Eulalie Besson b: 8 APR 1869 in Thibodaux, Lafourche Parish, Louisiana
Lydia Besson b: 1871 in Thibodaux, Lafourche Parish, Louisiana