觉 seems to have two pronunciations:
睡 觉 - shuìj iào - “to sleep” (DuChinese Cat 1 Chapter 1 story) vs
觉 - jué - “Senses” in HH
The HH doesn’t seem to take into consideration the " a nap / a sleep" meaning (HSK6) - I dont know why DuChinese chose an HSK6 char for one of their most basic stories
ChatGPT told me that about “20–30% of the commonly used characters are polyphonic” 
How does it work with the HH system, because it doesn’t seem to currently be included?
Great question!
As you’ve noted HanziHero only supports one pronunciation per character. This pronunciation is selected by how common it is, words included. The rationale is so one doesn’t have to worry about/get confused on multiple pronunciations as you get familiar with characters and some initial words. Some could find even the tone 5 switch ups on words confusing as it is!
Our tentative plan for alternate pronunciations within a character is to block them and ask for the current main one, to make sure one doesn’t later mix up pronunciations when going through words. In terms of getting familiar with the alternative pronunciations of a character, one would slowly grow familiar through more immersion.
There is more we could do here to minimize confusion/surprise and make it easier to learn the other pronunciations. Whether to maybe have a list of alternative pronunciations to alert you it is polyphonic, or having the blocking mechanism above 
For me an alternate pronunciation could be resolved in the simple way - which is already implemented in HH
The alternate pronunciations appear only in the certain words. Just teach these words with the modified mnemonics like you already do with the [5-silent] words.
For example:
[睡 觉]
[睡] - to sleep
[觉] - senses
[睡 觉 To sleep] means to be in [睡 sleeping state] and to turn off your regular [觉 senses], which are from this point your different [觉* dream senses]. In your dreams you sense a terrible stink of manure because [Jimmy Neutron] is building in the cow manure cannon in the [Bathroom] of the [Barn].
Or something like this 
Learning all the possible pronunciations of characters without the context given by the words have no sense because it does not work like that.
Even in English.
In the phrase Pacific Ocean the C is being pronounced in three different ways. But when you are teaching kids alphabet you don’t try to make them memorize all the possible pronunciations of “c” - it has to come with words.