One thing I see people say about dark side Force-users a lot is that physical pain, the kind of which is received from anything that pretty much harms you, makes them more powerful, and it is often used as a response to the excuses that "Kylo Ren was injured in the fight in The Force Awakens and that was why he lost to Rey".
In Legends (Mainly The Old Republic) they delve a bit deeper on the whole Jedi vs Sith ideology, but some aspects are also seen in the movies (mainly the prequels) But this is mainly theory(in the movies) seeing it's not blatantly stated .
They both have a code to abide by and it shows that they aren't Good VS Evil but more Order VS Chaos. The Jedi draw strength by being in control of their emotions while the Sith indulge themselves in their emotions. With the Jedi one seemingly stronger in defensive practices while the Sith one more offensive.
Fear is the path to the dark side. Fear leads to anger. Anger leads to hate. Hate leads to suffering. I sense much fear in you. -Yoda to Anakin
Fear, Anger and Hatred are all three emotions and seen as a gateway to the darkside. But one thing that Yoda doesn't mention is Love, another emotion, the only reason why Anakin has the fear of loss is because he loves his mother. Love (any form of attachment) is forbidden in the Jedi order due tot he emotions that come with it.
Anakin for example massacred an entire Tusken Raider encampment fueled by his hatred and grief. And yes tusken aren't the most powerful creatures in the galaxy but the sheer numbers should be a problem for somebody who was still at the rank of Padawan. Later he also managed to hold his ground against Count Dooku with the potential loss of Padme being on his mind fueling his strength. And in the rematch again his anger was showing (because Dooku cut of his hand) granting him more power then Obi-Wan who was several years his senior and considered a master.
But besides Anakin others have also shown this trait, Qui-gon and Obi-Wan fought Maul together and Maul was easily able to hold his ground and kill the Jedi Master. But suddenly Obi-Wan, who was only a Padawan, was able to hold his ground and eventually defeat Maul alone, it would be logical to say that the burst of emotions Obi-Wan felt when he saw his mentor die allowed him to tap in the the power of the dark side and defeat Maul.
So no it's (probably) not the pain that makes Kylo-Ren stronger but the hatred he feels because of the injury he had suffered. This is the same reason why Palpatine placed Vader in a less then ideal armor that was rather painful for his apprentice, because the constant pain made him angry and fueled his power. And later on when Vader betrayed his master he was able to draw strength for the love he felt for his son. The wound itself would have weakened him, making the logic that Rey beat him because of it plausible. But Kylo would have tried to compensate this disadvantage by drawing more power from his emotions.
The canon novel Lords of the Sith has the following passage with Vader meditating (emphasis added):
Without the neural connection to his armor, he was conscious of the stumps of his legs, the ruin of his arms, the perpetual pain in his flesh. He welcomed it. Pain fed his hate, and hate fed his strength. Once, as a Jedi, he had meditated to find peace. Now he meditated to sharpen the edges of his anger.
The concept of pain making dark side users more powerful also exists in Legends, perhaps most notably with Darth Sion, the "Lord of Pain", who used his pain from what should have been a mortal wound to fuel his hate and keep himself alive using the dark side of the Force.