J.L. Carr drew illustrations and cartoons for The Northants Campaigner, the bi-annual magazine of the Northamptonshire branch of the National Union of Teachers, which he edited and published between about 1963 and 1970. I haven't seen all issues, nor have I listed below all the illustrations as there are too many. I have just shown three examples.
Carr had first started trying to get his illustrations published immediately after the war, as he was probably trying to make some money to supplement his teacher's salary. After some searching, I found the only cartoon he drew for the magazine Punch, published in July 1947.
When he was working with London Magazine Editions to publish his fourth novel How Steeple Sinderby Wanderers Won the F.A. Cup he wrote in a letter to Alan Ross*:
Yes OK about sending review copies to Birmingham Post and Yorks Post . . . Mention to the Birmingham Post that I worked and lived many years in Birmingham and (I astonish myself) used to illustrate the weekly story for their Birmingham Weekly Post, now dead, and for their Sports Mail used to do a weekly drawing, a sort of cricket believe it or not. Practically worked for them you see! These were hard times in the Carr menage.
I looked through all issues of the Birmingham Weekly Post from July 1946, which is about when J.L. and Sally Carr moved to Birmingham after three months 'demob' leave, and November 1951, when the Carrs left Birmingham to live in Kettering.
I found two possible drawings by him, which are shown below. They were not done for the weekly short story, but for two special stories written by E. Baker. Later stories by E.Baker had a different illustrator.
I also looked though the Birmingham Mail Sports Final (a separate sports' newspaper published on Saturdays) from August 1946 to December 1951, and in the Birmingham Sports Argus from July 1947 to November 1951, but did not find any drawings about cricket that could be by Carr and were not signed by other artists. Wherever Carr's drawings were published, I suspect that they are similar to the drawing in the Midlands Club Cricket Conference Year Book, shown below. Carr had a very characteristic style of drawing people.
The illustrations and cartoons are shown in chronological order of publication.
*University of Leeeds Special Collections, ref. BC MS 20c London Magazine Editions 1/4/1/4
Source: Punch
Date: 16th July 1947
Details: Vol CCXIII, No 559
Page: 57
Title: none
Author: none
Signed: J.L. Carr
Notes: This is the only known cartoon by Carr published in Punch
Source: Birmingham Weekly Post
Date: 25th August 1951
Page: 4
Title: A Challenge Match on the Village Green
Author: E. Baker
Signed: no
Notes: This is not signed but the drawing is in Carr's style.
Source: Birmingham Weekly Post
Date: 8th September 1951
Page: 16
Title: "Skinner Gee" the Black Country Miser
Author: E. Baker
Signed: no
Notes: This is not signed but the drawing is in Carr's style.
Source: 1951 Midlands Club Cricket Conference Year Book
Date: 1951
Page: 23
Signed: J.L. CARR, lower left
Notes: This is printed on a single page with no caption.
Source: The Northants Campaigner
Date: Undated but probably Summer 1965
Page: Cover
Signed: no
Notes: This is an example of a drawing by Carr on the cover page of the magazine that he edited.
Source: The Northants Campaigner
Date: Undated but probably Summer 1966
Page: not numbered
Signed: no
Notes: This is an example of a full page cartoon in the teachers' magazine that he edited.
Source: The Northants Campaigner
Date: 1970
Page: [10]
Signed: no
Notes: This is an example of a drawing by Carr to illustrate an article in the last issue of the teachers' magazine that edited.