I really love how perfectly the show captured the essence of Maekar as a father, despite his very limited screen time and even more limited interactions with his sons.
Maekar is a harsh man, a fierce warrior, a loyal fourth son who's way more gifted than two of his older brothers yet always living in the shadow of his unimaginably charismatic eldest brother. His beloved wife has passed away, and it's rather obvious how deeply her loss sits with him (never sought a second marriage or even a relationship that we know of, even after he ascended the Iron Throne). He's a widowed father of six children, and all four of his sons are the kind of combo that would give most parents an aneurism. Despite that, it's made clear that even though he's far from a perfect father and struggling a lot with parenting, he loves his children fiercely.
Throughout the show, we mostly have evidence of his relationship with Aerion.
Aerion may as well be a psychopath or a sociopath, but it's obvious that he has been raised to be extremely pampered and spoiled too, which only adds to his dangerous tendencies.
That's a result of Maekar being unable to say no to him, most likely. Aerion also appears to be perfectly calm and unbothered even when Maekar is clearly furious at him, which shows that Maekar is an all-bark-no-bite father; the kind of parent that does chastise and scold his children, but never enacts any real punishment on them. Just think about it; even after the events of AKOTSK, the "punishment" Aerion receives is being sent to spend a few years in Lys, which is the Westerosi equivalent of Mykonos. That's not a punishment, that's a slap on the wrist that comes with a blow job (quite literally).
Raymun tells us that "Aerion is all smiles and chivalry so long as his father is watching." This is a behavior often attributed to psychopaths (they are often cery charismatic and excellent in acting the way people expect them to). I bet that Maekar has found it very hard over the years to believe everything people told him about Aerion's behaviour outside of his surveillance. In the show, I believe that we're at that stage where he's being forced to accept it.
Nevertheless, he still goes into a Trial of Seven to defend him, even though he very much recognizes Aerion is in the wrong. When Aerion is stabbed for the first time, he loses it to the point that he delivers what later proves to be a death blow to his own brother just to get to him, terrified that he's about to lose one of his boys. Despite everything that Aerion is, Maekar still loves him, deeply.
In Aegon's case, I think it's a given how Maekar cherishes him and how much hope he has for his little one. The thought of Egg being separated from him as Dunk suggests is just too unbearable. Plus he's the only Maekarling with whom papa shares a tender moment. No need to elaborate.
... and then there's Daeron (oh boy).
Daeron quickly became a favourite of mine in the show, and I think it's a real shame we didn't get any scenes with him and Maekar. All we have regarding their relationship is the dialogue between Baelor and Maekar upon their arrival at Ashford, in which Maekar says
(... ooof. I'm willing to bet this line haunts him for the rest of his days later on 😞)
Baelor later tells Dunk that "Maekar has placed all his best hopes on Aerion, since Daeron has been such a grave disappointment to him,” but frankly, as much as I love Baelor, I don't think he was right here.
Maekar is putting a great deal of pressure on Daeron to be a warrior like himself. As selfish and unreasonable and harmful as this is for Daeron, it shows that Maekar still views him as his future heir and truly believes that he can do better and make something truly worthy of his life - and that's why he's so frustrated with him. Daeron could have been the perfect Targaryen prince; he's young, handsome, kind, smart and funny, and in Maekar's eyes, he's wasting all of his gifts by drinking and whoring instead of training to be an heir. He has far from given up on Daeron - otherwise, he would have just accepted the situation and wouldn't be pressing him so hard to take part in tourneys, give up on alcohol etc.
Maekar might not be my favourite Targaryen, but I always thought him to be one of the most tragic ones. I know this is a Dunk & Egg show rather than a Targaryen one, but the writing on them was great and all the Targaryen actors did a phenomenal job of portraying the characters on screen, so I really hope we see more of this stressed father and his nightmares of sons in the future.
Plus, you know. Dilf problematic fathers who fiercely love and cherish their children are some of the hottest people around.