Depending on the style you're aiming for, just use the single letter.
For example, in Franz Kafka's novel The Trial (Der Prozess), the protagonist's name is Josef K. Throughout the novel, he is simply refereed to as K. - not just by the narrator, but also by other characters, who however mostly use the formal Mr. K. This is not unusual for Kafka - if I'm not mistaken, he does that in a few novels - but it should be noted that this gives a specific feel to the story and the relationship between reader and protagonist, which is perhaps not what you want.
Reading The Trial feels a bit like reading a report already, but the fact that character is just K. makes him more distant to the reader and emphasizes the report-like feeling. I don't want to spoil the end, but as is typical in Kafka's works (or at least the books I read), it doesn't end well for K. And although it's sad how it ends, I didn't really feel sad for the protagonist; his actual name isn't even known at the end.
Of course, this is subjective. Different readers get different things from stories and it's been ten years since I read it the last time. Because of that I recommend taking a look at The Trial.
Regarding sound and pronunciation
There is only so much you can do to give the reader the right idea how it is pronounced. In your example it is probably not even necessary since in English G is already pronounced exactly how you want it. However, if you want to be absolutely sure that the readers get how it is pronounced, because it is important to the story or it could get lost in translation*, you should simply mention it. Depending on the point of view and your style this could be done by the narrator
G didn't mind getting called just by his first initial, but he hated it when people mispronounced it just to mock him. Not a single pun he didn't already hear, and the get worse with time. Still G didn't grew tired of correcting people that it's pronounced just like the english letter, Gee.
or it could happen naturally in dialog i.e a character making a dismissive comment about G or a pun on G that only works if the pronunciation is wrong ( or right). Let the character correct them so the reader will know how it is pronounced. From that point on you can keep G in text, and the reader will still know its pronounced like the standard english letter G.
However this could be strange to read because it seems unnecessary to mention how G is pronounced. If it is not actually important to the story (because it has to rhyme or something) I would leave it to the reader.
*However if your story is not actually written in English then this might be of course important because even the same Latin letters are not pronounced the same in different languages. Take Kafkas K, for example. In German it would be pronounced Ka (with an A like in car) whereas in English it would be pronounced Kay (like in okay). I can't remember a single instance where this would make a difference in Kafkas novels but that doesn't necessary mean it's the same in your case.