While medieval poetry about knights is common, particularly in the stories of heroic deeds known as Chansons de Geste, the idea of the 'chivalric' knight, courtly and romantic, is largely a modern creation. Late medieval works like Mallory's La Morte d'Arthur populised the idea of the polymath knight, as skilled with the quill as the sword and given over to noble quests, but it should be remembered that these are aspirational works intended, like the Chansons de Geste to provide moral instruction and guidance to people who were often little more than well-born thugs.
That's not to say that knights or their equivalents couldn't be well educated - in the early medieval period, Alfred the Great mandates that all ealdormen, thegns and reeves should be capable of basic literacy and numeracy, and expected them to be familiar with certain books, and references to educated knights are common in High Medieval chronicles and histories - Orderic Vitalis records, for example, that William IX of Aquitaine had written a book of poetry before embarking on the Crusade of 1101.
Part of the problem with answering this question is, as u/BRIStoneman notes, that the “poet-knight” is far better attested as a medieval literary convention than as a concrete fact. The vast majority of named figures in the French and English literary traditions of the High and Late Middle Ages are usually professional or semi-professional poets (this seems to have been more common in France, which had a longer tradition of trouveres and poetic puys), courtiers of various stripes (nobility, diplomats, etc) and clerics. Even setting aside “proper” knights, the overlap between martial figures and poetic producers is seldom striking. Charles D’Orleans (1394-1465) was a duke and fought at Agincourt, but the majority of his poetic output seems to have occurred during or after his decades-long imprisonment (of the luxurious variety) in England. Geoffrey Chaucer (?1340-1400) was not noble but served as a page to the Duke of Clarence and fought at Rheims (and was captured and ransomed) in 1359. His poetic output also happened after his martial service, not during – his earliest work seems to have been written in the 1360’s, when he had begun his lifelong career as courtier, diplomat and civil servant.
There are, of course, some named authors who were knights in the strict sense – but I only have my knowledge from the later English tradition to draw from here. Sir Thomas Malory (c.1415-1471) wrote/translated/compiled the Mort D’Arthur, but again during a period of confinement rather than active martial service. It is, of course, a prose work – perhaps the most prosaic prose work in the whole Arthurian tradition (I confess that I am not a fan). A knight called Sir Richard Roos (c.1410-1481/2) translated Alain Chartier’s courtly love dialogue La Belle Dame Sans Merci into English poetry, but I confess I know very little about him. His biographer, Ethel Seaton (Sir Richard Roos: Lancastrian Poet, 1961) suggests that he wrote the poem in 1441/2, just before he actually became a King’s Knight and of course before the outbreak of the Wars of the Roses.
The most interesting poet-knight from my point of view is Sir John Clanvowe, author of The Book of Cupid (or The Cuckoo and the Nightingale). It actually shares some similarities with Roos’ poem, which you can find in the same edition. Clanvowe composed the poem during the late 1380’s, which was probably not a martially busy time for him, but he did fight soon afterward (he died on crusade in c.1390). It is hard to say exactly how typical Clanvowe’s poem might be as the product of a knight at court. He was famously part of a court circle (with allegedly Wycliffite tendencies) who were among the first to receive the works of Geoffrey Chaucer, and Chaucerian influence pervades the work. Several members of that coterie seem to have been sharing poetry amongst each other, and thereby shaping their literary tastes and habits. In its form and subject matter (courtly love lyric/complaint in the form of a dream vision, with a dialogic element) it is also very French, which is unsurprising for court poetry of the period. But it is very much a developed work (300 lines is neither especially long nor especially short for poetry of the period), which is probably why it has survived with a name attached, although the manuscript compilations in which it survives also suggest that the Chaucerian connection helped. I suppose what I’m driving at is what it isn’t - it isn’t a short, improvised lyric preserved by chance, nor a carefully prepared occasional poem for a specific event, nor a translation or other work requiring great specialist learning (although he did write a “Wyclffesque” prose treatise, The Two Ways, that attests to his education). The English court context is The Book of Cupid's major cultural influence, not his position as knight – one can comfortably imagine it being written by many contemporary English court figures, be they civil servant (Chaucer), lawyer (Gower), cleric (Lydgate) or royalty (James I).
This is where it starts to get tricky, because there’s little doubt that other knights in English court culture would have been writing poetry, even if they weren’t doing so “in the field”. Other knights in Chaucer and Clanvowe’s circle (and similar circles throughout the fifteenth century such as that which sustained John Lydgate) probably contributed as well. We know, for instance, that Sir Lewis Clifford supplied Chaucer with writings from the French poet Eustache Deschamps. But if work from other such knights survived then it probably does so anonymously, as much short English lyric does, in poetic compilations. There are plenty of reasons that such work might survive nameless, or not have survived at all: if it was too short; if it was attached to too little authorial prestige; if it was of low quality. A number of single-stanza Chaucer lyrics survive that we would totally ignore were his name not attached to them. It’s worth noting that the mode of circulation will have played a part too – bound books, particularly if they are seldom handled, survive far better than ephemeral single leaves, wax drafting tablets, and folded or loosely bound pamphlets. It is extremely rarely that we have an authorial holograph, still rarer a draft or fair copy. Poetry produced for the court will often never have left it.
To get back to the point: what can we safely assume about poetry by knights and other courtiers that failed to survive the test of time? The conventions of better-preserved courtly poetry suggest some likelihoods. It will have been either in French, or in an English that was heavily influenced by French style. Octosyllabic or decasyllabic rhyming couplets for longer poems, and lyric forms like the ballade, rondeau and virelai for shorter pieces. Probably in first person, either as monologue or dialogue. Some of it might have been sung. Even in English verse, new French loanwords and older loanwords that still retained French endings will have ameliorated the notorious scarcity of rhyme in English. Love will have been a popular theme – be it love anticipated, love achieved, or love lost. A Christian philosophical undercurrent may occasionally have reared its head in meditations on mutability and “true” love.
For an example, I’m going to cheat and use an inset lyric from Chaucer’s earliest confirmed poem, The Book of the Duchess (c.1368). Chaucer’s narrator encounters the so-called “Man in Black” (though it was once critical fashion to call him “The Black Knight”), who draws together the personae of chivalric knight, lover-poet, the recently widowed John of Gaunt, and Christ as Man of Sorrows. The “joke” of the Man in Black’s lyric “complaints” is that they rely so heavily on courtly poetic conventions that the narrator struggles to understand that the knight’s lady is really dead, not merely metaphorically distant. As an approximation of one of the kinds of lyric a knight at court might write, it is as good as any:
I have of sorwe so gret won
That joye gete I never non,
Now that I see my lady bryght,
Which I have loved with al my myght,
Is fro me ded and ys agoon.
Allas, deth, what ayleth the,
That thou noldest have taken me,
Whan thou toke my lady swete,
That was so fair, so fresh, so fre,
So good that men may wel se
Of al goodnesse she had no mete!
(I have so much sorrow that I never feel joy now that I see that my bright lady, who I have loved with all my might, is dead and gone from me. Alas, Death, what offends you that you would not have taken me, when you took my sweet lady? – she that was so fair, so young, so generous [or noble], so good that all can truly see that in goodness she had no equal?
Sources
Charles Muscatine, Chaucer and the French Tradition: A Study in Style and Meaning (Berkeley CA, 1957).
Richard Firth Green, Poets and Princepleasers: Literature and the English Court in the Late Middle Ages (Toronto, 1980).
Derek Pearsall, The Life of Geoffrey Chaucer: A Critical Biography (Oxford, 1992).
Ardis Butterfield, Poetry and Music in Medieval France: From Jean Renart to Guillame de Machaut (Cambridge, 2002).
---, The Familiar Enemy: Chaucer, Language and Nation in the Hundred Years War (Oxford, 2009).