Padecer means "to suffer" or "to endure." It's a more formal and profound word than simply sufrir.
In "La Puerta," the singer uses padecer to express a deep, prolonged suffering caused by lost love. He sings about trying to convince his beloved's soul of his padecer, highlighting the intensity of his emotional pain and the enduring nature of his heartbreak.
"La Puerta" is a classic bolero of heartbreak from Los Tres Ases, the vocal trio whose close three-part harmony helped define the genre in the mid-20th century. The song's central image says everything: "La puerta se cerró detrás de ti" [The door closed behind you] - a lover's exit rendered as one final, quiet sound, with nothing dramatic left to say about it.
What follows is the singer working through what came after. He'd believed he could win her back - "Creyendo que podría convencer / A tu alma de mi padecer" [Believing that I could convince / Your soul about my suffering] - but the relationship couldn't survive what life put in front of it. The line "La misma adversidad... nos vino a castigar con el dolor" [That same adversity... came to punish us with pain] frames the ending less as a betrayal than as two people worn down by circumstance. It's a song built on one closed door and an unfinished argument, sung with the unhurried harmony boleros are made for.