In a prologue to the later editions of The Lord of the Rings Tolkien writes
The most critical reader of all, myself, now finds many defects, minor and major, but being fortunately under no obligation either to review the book or to write it again, he will pass over these in silence, except one that has been noted by others: the book is too short.
Apart from being too short, do we know what other defects Tolkien is referring to?
Leaving aside the failures perceived by others, Tolkien's own criticisms are harder to find, but there are a few scattered throughout Letters:
Letter 131 may imply that he was unhappy with the thematic implications of Eowyn's unrequited love for Aragorn:
I will say no more, nor defend the theme of mistaken love seen in Eowyn and her first love for Aragorn.
The Letters of J.R.R. Tolkien 131: To Milton Waldman. 1951
It may also imply that he's simply tired of defending it against criticism; six and two threes, really.
Although really an error in The Hobbit, not The Lord of the Rings, I've discussed before that Tolkien came to regret naming a Troll "William", as well as a particular turn of phrase that seems to have caused confusion:
I might not (if The Hobbit had been more carefully written, and my world so much thought about 20 years ago) have used the expression 'poor little blighter', just as I should not have called the troll William.
The Letters of J.R.R. Tolkien 153: To Peter Hastings (Draft). September 1954
In Letter 154 he notes that the degree of modernity found in hobbit-civilization was probably a mistake (emphasis his):
Some of the modernities found among them (I think especially of umbrellas) are probably, I think certainly, a mistake, of the same order as their silly names, and tolerable with them only as a deliberate 'anglicization' to point the contrast between them and other peoples in the most familiar terms.
The Letters of J.R.R. Tolkien 154: To Naomi Mitchison. September 1954
He mentions in Letter 156 being dissatisfied with how Gandalf's return is narratively presented:
I think the way in which Gandalf's return is presented is a defect [...] That is partly due to the ever-present compulsions of narrative technique. He must return at that point, and such explanations of his survival as are explicitly set out must be given there – but the narrative is urgent, and must not be held up for elaborate discussions involving the whole 'mythological' setting.
The Letters of J.R.R. Tolkien 156: To Robert Murray, SJ. (draft). November 1954
In Letter 166, he expresses concern over how his handwriting is typeset:
The compositors always make mistakes in setting from my handwriting!
The Letters of J.R.R. Tolkien 166: To Allen & Unwin. July 1955
Although this letter was sent during the editing phase of Return of the King, and many of the errors he refers to were (probably) corrected, it seems likely that some errors of this kind slipped into the final published volume
In Letter 173, he expresses a desire to write something more about Sam and Elanor:
I still feel the picture incomplete without something on Samwise and Elanor, but I could not devise anything that would not have destroyed the ending, more than the hints (possibly sufficient) in the appendices.
The Letters of J.R.R. Tolkien 173: To Katherine Farrer. October 1955
In Letter 187, he laments having been forced to leave out his reproductions of pages from the Book of Mazarbul (emphasis his):
Reluctantly also I had to abandon, under pressure from the 'production department', the 'facsimiles' of the three pages of the Book of Mazarbul, burned tattered and blood-stained, which I had spent much time on producing or forging. Without them the opening of Book Two, ch. 5 (which was meant to have the facsimiles and a transcript alongside) is defective, and the Runes of the Appendices unnecessary.
The Letters of J.R.R. Tolkien 187: To H. Cotton Minchin (draft). April 1956
In Letter 211, he remarks on an error in Sam's invocation of Varda (emphasis his):
The use of O on II p. 339 is an error. Mine in fact, taken over from p. 338, where Gilthoniel O Elbereth is, of course, a quotation of I p. 88, which was a 'translation', English in all but proper names. Sam's invocation is, however, in pure Elvish and should have had A as in I p. 250. Since hobbit-language is represented as English, O could be defended as an inaccuracy of his own; but I do not propose to defend it.
The Letters of J.R.R. Tolkien 211: To Rhona Beare. October 1958
But this error was corrected for the second edition.
Later in Letter 211, he remarks that it was incorrect of him to describe Glorfindel's horse as wearing a bridle:
[B]ridle was casually and carelessly used for what I suppose should have been called a headstall. Or rather, since bit was added (I 221) long ago (Chapter I 12 was written very early) I had not considered the natural ways of elves with animals. Glorfindel's horse would have an ornamental headstall, carrying a plume, and with the straps studded with jewels and small bells; but Glor. would certainly not use a bit. I will change bridle and bit to headstall.
The Letters of J.R.R. Tolkien 211: To Rhona Beare. October 1958
One such reference was indeed removed in the second edition, but references to the bridle (and saddle, which Elves also don't require) remained in later editions; for example (emphasis mine):
Checking the horse to a walk, [Frodo] turned and looked back. The Riders seemed to sit upon their great steeds like threatening statues upon a hill, dark and solid, while all the woods and land about them receded as if into a mist. Suddenly he knew in his heart that they were silently commanding him to wait. Then at once fear and hatred awoke in him. His hand left the bridle and gripped the hilt of his sword, and with a red flash he drew it.
Fellowship of the Ring Book I Chapter 12: "Flight to the Ford"
He notes several typographical errors in the Puffin edition in Letter 236 (emphasis his):
I believe there is only one error remaining in the text from which the Puffin was printed: like for likes (6th imp. p. 85 line 1; Puffin p. 76, line 23). This crept in in the 6th imp. I think. Not that Gollum would miss the chance of a sibilant! Puffin has not emended it. I suppose Gollum was regarded as 'without the law' and immune from the dictates of dictionaries or 'house-rules'. Not so the narrator.
Apart from this the errors appear to be few. I have noted: waiting is omitted before/or (puffin p. 32/11). ahead appears as head (p. 87/5 from bottom). There is an inverted g in examining (p. 225/2 from bottom). And oubht, bood appear for ought and good on p. 228.
The Letters of J.R.R. Tolkien 236: To Rayner Unwin. December 1961
Tolkien uses “veranda” twice in The Hobbit. Once, in the narration, which is fine since it’s a part of Tolkien’s language. But he also has one of the Dwarves in The Hobbit say “veranda” referring to a porchlike structure on Beorn’s house.
Veranda is a Hindi word, so unless people of Middle-earth had trade routes with India—error—it should not be a part of Middle-earth speech. Tolkien should have had the Dwarf say “porch.” There is nothing in The Hobbit or LotR that even hints at India or Hindi, so supposing isn’t really a solution.
Up jumped Bilbo. “Breakfast!” he cried. “Where is breakfast?”
“Mostly inside us,” answered the other dwarves who were moving about the hall; “but what is left is out on the veranda. We have been about looking for Beorn ever since the sun got up; but there is no sign of him anywhere, though we found breakfast laid as soon as we went out.”