In Kingdom Come: Deliverance, it is claimed that many churches actually had relics of saints and other Christians, but many were fake. Did churches really claim to have holy relics, and if so, what were some of the most notorious examples?

Yes, relics were extremely common and extremely valued during the Middle Ages, and among Catholics well into the Early Modern era! (I mean, they still are valued by Catholics, just for slightly different reasons and in a different cultural context.) Many churches claimed relics; relics might be housed in elaborate vessels (reliquaries) and the reliquaries might get quite elaborate and lavish. Not all relics are purported to be the physical remains of saints (a bone, a hank of hair, or a bloodstained cloth, for instance) -- others are merely physical artifacts (like a vial of Mary's breastmilk, or the chains worn by St. Peter) associated with their respective saints and imbued with their power through that association. As trade in relics and traffic in pilgrims going to see those relics became more lucrative this provided an obvious incentive for relic-fakery, and it was already difficult to compare overlapping relics held between different churches on different sides of Europe. Canon law tried to govern the trade of relics and to assure some degree of authenticity in the form of certificates or letters verifying the relic's nature, but those could be forged as well, especially when relic-sellers trafficked to laypeople directly.

Chaucer's description of the Pardoner in the Canterbury Tales details one late 14th century imagining of how a person might go around dealing fake relics, not merely to institutions but to individuals:

But of his craft, fro berwyk into ware,

Ne was ther swich another pardoner

For in his male he hadde a pilwe-beer,

Which that he seyde was oure lady veyl:

He seyde he hadde a gobet of the seyl

That seint peter hadde, whan that he wente

Upon the see, til jhesu crist hym hente.

He hadde a croys of latoun ful of stones,

And in a glas he hadde pigges bones.

But with thise relikes, whan that he fond

A povre person dwellynge upon lond,

Upon a day he gat hym moore moneye

Than that the person gat in monthes tweye;

And thus, with feyned flaterye and japes,

He made the person and the peple his apes.

The pardoner passes off a "pilwe-beer", a pillow case, for Mary's veil; he passes off a shred of material for a piece of St. Peter's sail, a brass cross set with stones like gold set with jewels, and pig's bones under glass. This might be merely a bottle he's lugging them in, but it might be something more sinister in this context -- people liked to see what was inside the reliquary, to be able to peep through a glass or crystal window and see the material inside, so he's not just selling them a pig in a poke, so to speak, he's taking the deception to the next level. And he sells these things, not just to churches but to poor people, for the equivalent of two months' wages at a pop. Real-life relic forgery was a potentially elaborate and lucrative enterprise, but you can see here how it's not viewed in a charitable light -- the Pardoner is a skeezy creep. (The whole pilgrimage is to go worship at the church holding Thomas Becket's relics in Canterbury.)

Of course, in modern terms a lot of these saints' remains or artifacts would be of impossible-to-prove provenance -- even if the fingerbone under a crystal door in a gold reliquary shaped like a hand turns out to be a human fingerbone of a certain age, how would one go about proving that it belonged to a specific martyr or a specific saint? The main reason why contemporaries deduced relics were "fake" was when there was simply too many finite objects to go around -- it's one thing to have hundreds or thousands of people claim they're in possession of shards of various sizes of the cross Jesus was crucified on, because that object is divisible, but a hundred crucifixion nails, a hundred Jesus foreskins, or a dozen shinbones of St. So-And-So? These seemingly mutually-incompatible relics began to elicit scrutiny within the Church and without as far back as the 10th century, and their very existence was so numerous as to have become notorious; it was one aspect of Catholic worship that later came under Protestant criticism during the Reformation. For instance, John Calvin's writing regarding relics was pretty frank about the number of relics exceeding all possibility, on top of relic use in worship being potentially tantamount to idolatry; he remarked that there were enough pieces of the supposed "true cross" floating around Europe that one could build a ship from them, and enough vials supposedly containing Mary's breastmilk that the Holy Virgin must have been a cow.

The second part of this work [Traité Des Reliques] consists of a vast panorama of holy relics throughout catholic Europe in the XVIth century. Calvin says that it is indeed a deep “forest”, from which it is difficult to find one’s way out ; this is why he recommends a methodical approach. First, concerning Christ : his teeth, hair, blood, cradle, the column against which he leant while arguing in the Temple, the wine jars of Cana, the bread, the cross, the shroud, Judas’ deniers, the crucified body with the beginnings of a beard or weeping eyes. Next, the Virgin Mary : her bones, hair, milk, clothing, belt, slipper and hair clips. Then later, Calvin goes on to speak about the saints who were with Jesus : Saint John the Baptist and pieces of his head, his jaw, his ear, his finger ; Saint Peter and his body, his head, his teeth, his brain, his episcopal pulpit ; saint Paul and his body, his shoulder, his bones, the chains with which he was held. Finally, Calvin writes more summarily about the relics of other saints : Thomas, Denis, Etienne, Laurent, Gervais, Protais, Pétronelle, Hélène, Hilaire, Honorat, Symphorien, etc. Several of them have so many relics that according to Calvin, it would be possible to reconstitute several human bodies ; Andrew, the two James’s, Philip, Simon, Ann, Lazarus, Madeleine, Sebastian, Anthony, etc.

Paul-Alexis Mallet

The most notorious relic in modern terms would be the Shroud of Turin, purportedly a shroud imprinted with Christ's image; the most famous in medieval terms would perhaps be the relics of St. James the Great at Santiago de Compostela, St. Peter's relics in Rome, or the relics of Becket at Canterbury.