Here are my suggestions(if you care to take them)/examples.
" “Look, she- my friend- told me that Icarus is just an alibi…” " - I think you mean alias? I just noticed this and thought I’d point it out.
" The amber gaslight dances on particles of dust that float around you. The usually enticing swirling motion, the odd waltz of rust and dirt, fails to enchant you with its normally mystic properties. " - I might replace the second ‘the’ with ’ It’s ', and the third ‘the’ with ‘an’.
" You see the coal leave a black smudge on the clay cover and you use the back of your shirt to try and rub it off. " - sorry if you don’t want grammar corrections, but I think there should be a comma between ‘cover’ and ‘and’.
" All it does is smudge the ugly spot until it mars the edge of the elegant golden title. The title you can’t read but know what it says. " - Using the word title here so close together seems a little repetitive. If you want a suggestion, I would just move ‘the title’ to be after ‘you can’t read’, just so they’re a little further apart. I would also add a ‘you’ after ‘but’, and a comma before ‘but’.
" Rather, they weren’t answering the door, lest it was a ghost that nearly hit you on the head with their book. " - ‘Lest’ in the place of ‘unless’.
" Even though you can speak just fine, you’ve never heard anyone speak those two words. " - I would probably replace the second ‘speak’ with ‘say’.
" The City, this underground behemoth you’ve lived in all twelve years of your life. The City is alive, and you would know- you can talk to it. " - I would probably combine these two sentences by getting rid of the second ‘the city’ and just putting a comma between ‘life’ and ‘is’, or I would replace the second ‘the city’ with ‘It’ if you want to keep the two sentences seperate. In either case I would also replace ‘this’ with ‘an’.
I think that, in general, you use ‘the city’ way too often. I would probably replace that with ‘it’ or some other word when it’s possible (and fitting), to keep the phrase from becoming too repetitive and boring. Like here: ‘…The City, who replaced your faulty fleshy parts with pieces of The City. Making you half-city and half-human.’ you might change the second ‘the city’ into ‘itself’ instead. I would also get rid of the first period and just combine the two sentences into one with a comma, because ‘Making you half-city and half-human’ doesn’t really work on its own.
" Your left leg is metal from the knee-down, it can stiffen up when you walk, so you have to make sure it’s well oiled. " - I think that here you should actually seperate ‘Your left leg is metal from the knee-down’ and ‘it can stiffen up when you walk, so you have to make sure it’s well oiled’.
" You remove the leg when you sleep. The leg itself isn’t as flexible as your skin-covered leg, but it’s stronger. You were raised by The City, and your mentor. Those two are the only people you’ve ever known as “parents”. " - I’m not sure what suggestions to give here, other than “find a way to make it flow better”. It sounds very halting, like you’re just listing off random information.
" You wouldn’t know, of course, but you’re familiar with the shape and sound of your name, even if you may never know exactly what each symbol means, or how they form the sound that you and others use to call you. " - To me it seems a bit redundant to say ‘you wouldn’t know’ and then also say ‘you may never know’. I would probably get rid of 'You wouldn’t know, of course, but ’ and just have the sentence start off at ‘You’re’.
" You lack a surname, as the surname is usually a marker of a human family. Yet your family is The City itself, and your surname may as well be your left leg, and the heavy thud you make whenever your metal foot touches the ground beneath you. " - More of a personal preference, but I’d probably rewrite this sentence as ‘You lack a surname, as that is usually a marker of a human family, and your family is The City itself. Your surname may as well be your left leg, and the heavy thud you make whenever your metal foot touches the ground beneath you.’
" But it’s not as if you need one, even if you were older you’d simply be addressed as “City Child”, never Mister Ovid. " - I might rewrite this as ‘But it’s not as if you need a surname. Even if you were older you’d simply be addressed as “City Child”, never Mister Ovid.’
" So it’s never bothered you, mainly because it’s never had to. " - This sentence just doesn’t really make sense to me. Your lack of a surname has never bothered you because it never had to? It might make more sense as ‘mainly because you’ve never had a reason to want/need a last name’, or something similar.
Some of your simple sentences could easily be combined to, in my opinion, give the text a better flow. I’ve already listed a few, but I’ll give one more here (and not list any more, because I don’t want this list to have a bunch of repeat suggestions).
Again, I don’t want to give too many repeat suggestions so I’ll just leave some general suggestions here:
- If you re-read a paragraph or page and find that you’re using the same word to begin a lot of sentences, you should change the sentences around a bit (ex. " You stand from the corner you’d been wedged in, carefully grabbing the book and holding it against your chest. You walk to the edge of the platform, and peer through the mesh-wiring at the sprawling City below you, past the spider web of rusted piping. You see people move like ants through the City’s veins, and double check that there’s no leaking- that was what you were originally sent here to do after all- before heading for the staircase. ", a whole lot of you’s).
- You should re-read a paragraph or page (or have someone else read it for you) to find when you’re repeating words in general, because that does seem to be a problem you have (and that’s a shame because I think most of your writing it very good otherwise).
- Have someone else read a paragraph/page you’ve written (or leave it alone and come back after a while) to find any sentences or words that seem redundant or useless/unnecessary (the ‘while’ in “While most City’s Children know their mentors by name and face.” has no purpose, for instance, and “Mentor listens while pouring your soup, but you can tell by the swing of the hood that they glance at you now and then, assuring you that they’re still listening.” mentions that your mentor it listening twice, which is redundant).
Sorry if this is more than/not what you wanted! I didn’t really mean to go through every sentence I had the slightest nitpick with, but I couldn’t help myself 
So, I stopped after a few pages because I didn’t want the list to get too long. Most of there are kinda minor (and, like I said, nitpicks), so I think I may have veered off of what you wanted for examples. These are just suggestions of how I think you could improve the flow a bit more, and of course you don’t have to take my advice. Sorry it’s so long, I really didn’t mean for it to be.
Also: Yes! Removing most of the albeits did make it flow much better, and I was kind of confused at first because I didn’t know that you’d removed them and I thought that I was just going crazy 
