Interesting question. In the Englishes spoken in the USA, "h" on its own is not ever silent. Some Englishes spoken in English seem not to pronounce any "h" letter In general, "h" when it is preceded or followed by a vowel is not silent. Think of the word "he". Now say the word "he" repeatedly and notice the constriction at the top of your throat and the position of medial and distal edges of your tongue pressing against your upper palette. Push the air out from your abdomen to make the sound. In general "h" is a single sound that is made in the way just described.
When "h" is in combination with a preceding consonant, the duo or "dipthong" have a special sound as a unit. Examples of diphthongs with "h" include:
THIS SOUNDS LIKE "sh" (Chevrolet), "k" (Buchanan), or "?" (church, punch, purchase).
dh and gh and kh appear in words that are an foreign that are using the English alphabet to represent sounds in another language. Here are 3 examples that come to mind: dharma, ghee, Khatereh.
gh is silent in words such as thorough, through, bouoght, bough
THIS SOUNDS LIKE "f"
THIS IS A NORMAL "H" but the "h" sound may get hidden in such words as rhinoceros and Rhodes. So if you'd like you may consider the "h" in the "rh" combination followed by a vowel to be a silent "h" sh
THIS SOUNDS LIKE signalling to be quiet, not not make noise.
th THIS HAS 2 SOUNDS, you learn to pronounce each word individually: voiced th and unvoiced th. Some non.English speaking countries may use "th" to represent "t". For example in Mexico, "Martha" is the spelling used to represent this sound "MAR ta".
This too, like rh, when followed by a vowel can be treated as if the "h" is silent.
The next book that you read in English, use an audio book as you follow along with the written text. One reason, I suspect, is because of learning English through written words more than from heard/spoken words.