Alrighty! So! Here's the critique you asked for. I also went over your story and made a few edits and suggestions which will be below.
Starting with critiques, I firstly have to compliment your writing as a whole. You actually have a really solid style and great understanding of making a story flow smoothly. Most of my critiques and edits are just small things to keep in mind and to add that extra bit of smoothness (mostly from experience and learning what reads a touch better). You do a great job staying away from passive voice and making each sentence expressive. You used one adverb, which is impressive. Generally, reword sentences over using adverbs in the writing world, but personally, I think they're fine.
Overall, I don't really have much to critique or change. Most I can suggest outside the edits below is to now start leaning into setting. There isn't much, and you can really make a story pop by helping visualize the area.
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Here are my edits:
"I can't believe it."
Tala rolled her eyes. "Again with this?" she asked irritatedly in irritation. "It's been an entire day since it's happened. For the last time, FORGET IT."
"I simply can't!" Tyr whined. "You try having something that embarrassing happen to you! You'll never let go of it."
Tala rolled her eyes (Consider her squinting or turning away in exasperation. She just rolled her eyes a few sentences before). "Look Tyr, we all make mistakes. That's just life. You live with it and move on. Someday you'll laugh when you remember it."
Tyr, who had been laying on the ground, moping around the whole day, jumped (“sprang” is a vibrant word to consider instead of jumped)to his feet and huffed, "I don't want to hear your little lessons on life. I'm going for a walk."
"It'll be good for you," Tala called out as Tyr left their den.
Lyra looked at Tala quizzically. It had only been a few days since she had joined Tala and Tyr's pack, named Mazie : The Mazie Pack. They were a small pack, just starting out. It was only them three the three of them, plus two pups: Lesia and Alita, who had also joined quite recently.
Tala sighed. "Lyra," she explained to the confused, older wolf, "before you joined our pack - wait, actually," she mused, "before even the pups joined, Tyr went border patrolling for the first time."
“Lyra," she explained to the confused, older wolf, "before you joined our pack…" She paused and mused for a moment before continuing. "Wait, actually, before even the pups joined, Tyr went border patrolling for the first time."
Lyra nodded. "Ah yes," she said wistfully. "I always wanted to go border patrolling - it was a special privilege, you see, only for the best of the best. That was in my previous pack, where I grew up in."
"Well, consider yourself hired," Tala said with a smirk. "I doubt Tyr wants to do it anymore, considering what happened to him."
Lyra cocked her head. "Yes? Was that what he was complaining about so loudly?"
Tala nodded and grinned. "So here's what happened: he went border patrolling, right? And when he gets there he sees this rabbit, a singular rabbit. He thinks 'Oh, I'm a wolf. That's a bunny. Wolves hunt bunnies. Bunnies are weak compared to wolves,' right? So what happened, naturally is that he tries to fight the bunny, the bunny bites him, he misses, and it flees. So when he comes home an hour later, he's injured and there he has nothing for him to show for it."
Lyra raised an eyebrow. "So what I'm hearing is that Tyr got beat by a bunny."
"Yep."
Lyra whistled. "Wow, that really sucks for his ego."
"Yep"
Lyra smiled. "It's ok, I remember what happened to one of my previous pack's leaders. She was doing a PvP and -"
"Guys, I'm back!" Tyr interrupted, bursting back into their den. He smiled broadly and threw something down into the middle of the dirt floor. It was a dead rabbit.
Summer 
Night Weather: 
Earthquake Moon: 
