Count Dooku may have been one of the reasons why Yoda was hesitant to train Luke Skywalker in the Star Wars movies. Introduced in Star Wars: Episode II - Attack of the Clones, Dooku was Yoda's apprentice during his time in the Jedi Order. In the prequels, it's established Dooku left the Jedi after becoming disillusioned with their ways, and he went on to become a leader of the Separatist movement that led to the Clone Wars. During this time, Dooku was studying under Darth Sidious, unknowingly a placeholder while the dark lord moved in on Anakin Skywalker.
While Yoda had a hand in teaching all Jedi younglings the ways of the Force, onscreen he wouldn't have another direct apprentice until Luke Skywalker arrived on Dagobah in The Empire Strikes Back. Upon meeting Yoda, Luke was enthusiastic about continuing his training, but Yoda was initially reluctant, citing Luke's lack of patience among the reasons. Looking back at Yoda's personal history, it's easy to see why he didn't jump at the opportunity to take another student under his wing.
Star Wars canon has confirmed Yoda wanted to train Leia as a Jedi, believing she was the Skywalker twin best suited for the job. Yoda was disappointed when Obi-Wan requested he train Luke, feeling Luke had too much of Anakin Skywalker in him. But Yoda's stance may have run deeper than just noticing parallels between father and son. It couldn't have been easy for Yoda to see Count Dooku, a person he trained and bonded with, aligning himself with the ultimate evil in the galaxy, spurring events that caused the Jedi's downfall.
Yoda's desire to teach Leia suggests he was up for taking another apprentice, but it would have to be the right person. Luke's similarities to a younger Anakin were cause for concern for Yoda, as they made Luke more susceptible to turning to the dark side. Yoda's failure with Dooku likely exacerbated these feelings, and he could have been afraid that he wouldn't be able to help keep Luke on the proper path. In is exile on Dagobah, far away from the Empire's watch, the last thing Yoda wanted was to be responsible for the creation of another threat - this time one who potentially could be more powerful due to his bloodline. It's rare to think of Yoda as someone who doubts his own abilities and judgment, but after everything he'd been through prior to The Empire Strikes Back, it's understandable why he refused to train Luke at first.
Skepticism of teaching another apprentice seems to be a running trope in the Star Wars franchise. In Star Wars: The Last Jedi, the tables were turned on Luke as he would not train an eager Rey. The main catalyst for Luke's mindset, of course, was his own failure with Ben Solo, who had turned to the dark side and became Kylo Ren. Believing the galaxy was better off without the Jedi, Luke didn't want to run the risk of repeating his past mistakes. He even pointed out parallels between Rey and Ben's power, which made him afraid. Funnily enough, both Yoda and the elder Luke were proven wrong, but in the moment, their hesitancy arguably made some sense.
