I understand the explanation for “gu-” + “-eng” = “gong” but I just came across the following which - even after reading the above - I don’t understand (unless I’m missing something!).
日 is defined as Robin Hood (r) and House (-_) but where did the “i” go ?
For “I” specifically, we don’t have a final. There are a variety of reasons for this, but the main one is because “i” in Pinyin represents three distinct pronunciations, so wouldn’t fit easily into our mnemonic system of each letter groups (i.e., “Sound”) having its own distinct pronunciation.
These three separate pronunciations are actually represented in this excellent Standard Pinyin chart. I’ll go over each of them.
“i” can be pronounced like “ee” in the English word “see”.
“i” in Pinyin can also be pronounced like the “i” in the English word “sit”.
Lastly, “i” can be pronounced sort of like “i” in the English word “chirp”.
To get around the fact that “i” represents three distinct pronunciations (another of Pinyin’s MANY quirks), we make two small changes.
For the “ee” pronunciation, we simply create a new set of initials. E.g., “bi”, “di”, “ti”. This is useful because that specific “i” is also a medial in some cases (e.g., “bian” is “bi-” + “an”). It is similar to what we did with making a new set of initials with the “u” medial as mentioned in the post this thread was split from.
For the other two “i” pronunciations, we represent it by the null sound. This is the best of all choices. We can’t have a separate “i” final because it could still represent two different pronunciations. If we make a new set of initials (“si”, “zhi”, etc), those aren’t useful because the other two “i” pronunciations are not medials, so these initials could only be combined with the null final anyway.
Because of this, “ri” is represented as “r-” + “-_”.
All of that is quite a bit. I’m happy to go into more details about any of it to make it more clear. The gist of it is that Pinyin has major flaws that don’t work with a nice mnemonic system, and this modification works around those flaws.
As another note, all of this is much more clear in Zhuyin, where each distinct phonetic sound always has the its own symbol unlike pinyin where a group of characters can represent multiple different sounds. I don’t really use Zhuyin much at all, and often forget the symbols, but I find their breakdown to make way more sense.
For the “i” as “ee” in "see:
For the “i” in “sit” ones:
For the “i” in chirp ones:
Just a quick question to check my understanding - the above list set out doesn’t include ji, xi and qi but I assume that was just an accidental omission when writing out the above list ? Thanks! 
Yep, it was. The “i” in those syllables are also pronounced like the “ee” in “see”.