It looks like Hagrid's wand was not fixed. Hagrid's spells normally did not work out well.
Hagrid looked down at his umbrella and stroked his beard.
“Shouldn’ta lost me temper,” he said ruefully, “but it didn’t work anyway. Meant ter turn him into a pig, but I suppose he was so much like a pig anyway there wasn’t much left ter do.”
Harry Potter and the Sorcerer's Stone, Chapter 4, The Keeper of the Keys
Broken wand can be used for casting spells, but there is no guarantee that it will work. In fact, Hagrid does mention that he still has pieces of his wand and the fact that he uses his umbrella for performing spell is an indication that the broken pieces of the wand is inside his umbrella.
“Good wand, that one. But I suppose they snapped it in half when you got expelled?” said Mr. Ollivander, suddenly stern.
“Er — yes, they did, yes,” said Hagrid, shuffling his feet. “I’ve still got the pieces, though,” he added brightly.
“But you don’t use them?” said Mr. Ollivander sharply.
“Oh, no, sir,” said Hagrid quickly. Harry noticed he gripped his pink umbrella very tightly as he spoke.
Harry Potter and the Sorcerer's Stone, Chapter 5, Diagon Alley
We see Ron using his broken wand through out Chamber of Secrets with limited success.
“Ron was having far worse problems. He had patched up his wand with some borrowed Spellotape, but it seemed to be damaged beyond repair. It kept crackling and sparking at odd moments, and every time Ron tried to transfigure his beetle it engulfed him in thick gray smoke that smelled of rotten eggs. Unable to see what he was doing, Ron accidentally squashed his beetle with his elbow and had to ask for a new one. Professor McGonagall wasn’t pleased.”
Harry Potter and The Chamber of Secrets, Chapter 6, Gilderoy Lockhart