Carr acted as a reviewer for The Spectator between 1986 and 1993, I think. The search engine on the web site of The Spectator is terrible because a search on 'J L Carr' doesn't list any reviews. I found some by using Google to search The Spectator web site and found some more when I bought a collection of most issues of the magazine for 1987 and 1988. There may be more, perhaps earlier than 1986. I have the issues marked with an asterisk. If you know of more issues that contain a review by Carr, please let me know.
I have found only one review for another publication, The Guardian. I no longer have access to the archives of The Times and never had access to The Daily Telegraph. If you do, perhaps you would search them for me again, let me know.
J.L. Carr (1986). Laker's name outlives the man
. Cape Summer and the Australians in England by Alan Ross, published by Constable. The Spectator, 10 May, p 38-39.
J.L. Carr (1986). You behind the beard
. William Golding, the man and his books: a tribute on his 75th Birthday edited by John Carey, published by Faber & Faber. The Spectator, 27 September, p 28.
J.L. Carr (1987). A shadowy destabilising figure
. Jottings of a Genius by Dennis Peck, published by The Woodstock Press. The Spectator, 31 January, vol 258, no 8273, p 30.*
J.L. Carr (1987). Coming to life in his coffin
. Checkov by Henri Troyat, translated by Michael Heim, published by Macmillan. The Spectator, 11 April, vol 258, no. 8263, p 31-33.*
J.L. Carr (1987). Chewing off more than he could bite
. The Complete Notebooks of Henry James by Leon Edel and Lyall Powers, published by Oxford. The Spectator, 23 May, vol 258, no. 8289, p 53-4.*
J.L. Carr (1987). After the ball was over . .
.The Book and the Brotherhood by Iris Murdoch, published by Chatto & Windus. The Spectator, 12 September, vol 259, no. 8304, p 34-35.*
J.L. Carr (1987). Books of the Year. The Spectator, 28 November, vol. 259, no. 8316, p 34.*
J.L. Carr (1987). How to become a literary chap
. For Love and Money by Jonathan Raban, published by Collins Harvall. The Spectator, Christmas double issue 19/26 December, vol 259, no. 8319/8320, p 62 .*
J.L. Carr (1988). Xanadu in Northampton
. Penfriends from Porlock by A. N. Wilson, published by Hamish Hamilton. The Spectator, 27 February, vol 260, no. 8329, p 29-30.*
J.L. Carr (1988). The complaints of a violent family
. Out of this World by Graham Swift, published by Viking. The Spectator, 12 March, vol 260, no. 8331, p 28.*
J.L. Carr (1988). Strange doings in the capital
. The Newest London Spy by Tim Heald, published by Muller. The Spectator, 30 July, vol 261, no. 8351, p 29-30.*
J.L. Carr (1988). Books of the Year. The Spectator, 26 November, vol. 261, no. 8368, p 40.*
J.L. Carr (1989). Reports from another species
. Childhood: an Anthology edited by Penelope Hughes-Hallett, published by Collins. The Spectator, 28 January, p 28-29.*
J.L. Carr (1989). Perfect pitch. Cricket Cartoons and Caricatures, edited by George Plumptre, published by Collins; and The Essentialist by John Arlott, published by David Allen. The Guardian, 28 September.
J.L. Carr (1990). The mysterious benefactor from Shropshire
.The Potter's Field by Ellis Peters, published by Hodder Headline. The Spectator, 7 April, p 37.*
J.L. Carr (1990). Put it down in writing, Dad.
The Maker's Mark by Roy Hattersley, published by Macmillan. The Spectator, 30 June, p 40.*
J.L. Carr (1991). When the singing has to stop
. Jaunting through Ireland by Roy Kerridge, published by Michael Joseph. The Spectator, 23 March, p 37.*
J.L. Carr (1991). With never a stain, the pavilion of Heaven is bare
. By His own Hand: a study of Cricket Suicides by David Frith, published by Stanley Paul. The Spectator, 29 June, p 27.*
J.L. Carr (1991). Christmas Books The Spectator, 30 November, p 40.*
J.L. Carr (1992). You have not delighted us long enough
. The Faber Book of Socccer edited by Ian Hamilton, published by Faber. The Spectator 7 March, p 27-28. *
J.L. Carr (1992). Old fashioned war in Burma
. Quartered Safe Out Here by George Macdonald Fraser, published by Harvill. The Spectator 27 June, p 30-31.*
J.L. Carr (1992). The great Arsenal of democracy
. Fever Pitch by Nick Hornby, published by Gollancz. The Spectator 31 October, p 40.*
J. L. Carr (1993). If you go down to the woods today
. E.M. Forster: Interviews and Recollections edited by J. H. Snape, published by St Martin's Press. The Spectator 24 April, p 40-41.