First off, Martian Manhunter is my favorite superhero. He's one of the most complex characters to evolve from DC comics. He's a founding member of Justice League of America, and contrary to a popular believe, he's not a ripoff of Superman. Actually when he allows himself to bypass his morals, he's stronger. So how does he fit in with Supergirl? That's where I'm conflicted.
The show has gone a slightly different origin story, making Hank Henshaw his human identity and having him be Director of the D.E.O. ( Department of Extra-Normal Operations). That's not really an issue for me, since he's a shapeshifter and said he's been on Earth for 50 years already. It's honestly a fitting position for J'onn Jonzz, based on his history trying to preserve Earth from what occurred on Mars.
Supergirl has an entire episode, "Strange Visitor From Another Planet," revolving around J'onn J'onzz. They delve into the White Martian holocaust like extinction, and his psyche as a result of this. Since all Martians are telepathically linked, he's afraid to use any ability of his true form, i.e. outside of Hank Henshaw, using his powers will act as a beacon, driving the White Martians to Earth and possibly leading them to destroy his new family and home. That's not an issue either, in fact, it's very true to the core of the character.
Finally, he relates well with Supergirl emotionally, even better than Superman. Superman was raised human, and eventually learned he was an alien through the growth of his abilities. Sure he has to deal with the loss of his family like his compatriots, but at least he has Kara and was raised not knowing them. Both Kara and J'onn had their family stripped away while they were cognizant. J'onn has the added pain of surviving after fully witnessing his children getting incinerated. These are feelings and realities that no human can really understand.
My biggest conflict is how they weakened him, and perhaps it's too early in the development of the show, but it remains a point of emphasis. In all of these battles, he has his reservations to use his powers, but it still comes off strange. The character is being written as though he hasn't met Superman, or that he is new to Earth, yet they contradict that. Martian Manhunter has countless abilities of shapeshifting, telepathy, telekinesis, super-strength, flight and so forth. Usually, he has good reasons whether it be morals, or keeping his human identity in tack, but I guess what makes me twitch is the fact he still seems physically weak in the action. In many instances Supergirl or Alex needed to save him. In an episode early on in his reveal, he says Kryptonians are immune to his telepathy, which makes Superman quite happy. That is not canon to his character and is a shortcut for the plot.
I understand the show is about Supergirl. She's the star and she deserves it. I just have to question Martian Manhunter's role. I don't want him weakened. His character has already been slighted by not being in the Justice League film. He's too strong to be Supergirl's helper full time and he has the range and complexities to carry his own material.