Given the age difference between Henry and Delia, it's very likely he had a prior wife.

— born in Illinois:
4.  Israel H. CLAXTON, b. 1856/7

1.  Illinois Statewide Marriage Index, 1763–1900:

Follett, H W C	Claxton, Thursa	3 Nov 1872	Book D, p. 71	De Kalb
Fox, Robert J C	Claxton, Annie L	22 Dec 1874	Book E, p. 62	De Kalb

Who is Thursa?  These two, plus the marriages of Horace and David, are the only CLAXTON marriages in DeKalb County in this database.

2.  1850 Census Index/Images (online at Ancestry.com):  not found.  The only Henry CLAXTON indexed in the 1850 census is age 29 (b. TN, wife Nancy) in Hancock Co., TN.

3.  1860 Census Index/Images (online at Genealogy.com, Image #11 of 16):  Malta P.O., Malta Twp., DeKalb Co., IL, Roll 173 (Book 1), p. 442, PN 442, 3379/81, enumerated 21 Aug 1860, official enumeration date 1 Jun 1860 (extracted by Diana Gale Matthiesen):

Henry Claxton	53	M		Day Laborer	800	400	Eng
Delia     Do	38	F		House Keeper			Do
Horace    Do	16	M		Laborer			Do
David     Do	14	M					Do
Ann       Do	11	F					Do
Israel H  Do	4	M					Ill
Elisabeth Davey	83	F		Retired			Eng

Elizabeth DAVEY is almost certainly Delia's mother.

4.  1870 Census Index/Images (online at Genealogy.com, Image #27 of 32):  Malta P.O., Malta, [Malta Twp.,] DeKalb Co., IL, p. 460A, PN 26/27, 183/180, enumerated 12 Sep 1870, official enumeration date 1 Jun 1870 (extracted by Diana Gale Matthiesen):

Claton Henry	66	M	W	Ditcher	1000	275	Eng
______ Adelia	46	F	W	Keeping House			"
______ Horace	26	M	W	Engineer			"
______ David	24	M	W	Carpenter			"
______ Anna	20	F	W	At Home			"
______ Israel	13	M	W	"   "			Ills
Daire Edith	8	F	W	At School			"

5.  1880 Census (indexed at FamilySearch.org; page image online at Ancestry.com, Image #5 of 27):  Malta, [Malta Twp.,] DeKalb Co., IL, Roll 202 (Book 1), p. 113A, PN 5, SD 2, ED 33, enumerated 9 Jun 1880, official enumeration date 1 Jun 1880 (extracted by Diana Gale Matthiesen):

1880:  for an explanation of the column headings, please see What the Numbers in the Federal Census Mean (missing columns contained no data).
1	2	3	4	5	6	8	9	10	13	24	25	26
41	41	Claxton Henry	W	M	75			/	Laborer	Eng	Eng	Eng
		_______ Delia	W	F	56	Wife		/	Keeping House	"	"	"
		_______ Isreal	W	M	23	Son	/		Butcher	Ills	"	"

6.  1890 Census:  the 1890 Census Population Schedules were destroyed.

7.  1900 Census Index/Images (online at Genealogy.com):  not found (nor Isreal).  The only CLAXTONs in DeKalb Co., IL, were our subjects' sons, Horace and David.

8.  Messages posted to the CLAXTON Surname Board (online at Ancestry.com).