What is proper use of a True Random Number Generator? Let's say I have a true random number generator TRNG. Random numbers are generated on demand from a HW-entropy source, resulting in a 32-bit random value each time it is called.

I have to fill an array of 10 Bytes (=10*8 bits= 80 bits) with random numbers. Can I make a first call to TRNG, take the fist byte, the second, the third and the fourth to fill my array, and then call again...

Or should I call a new random number for each byte and throw away the non used bytes?

I would say the first approach is correct, but there is a slight issue with random number quality in my project and I want to be sure - maybe I'm completely wrong...
I assume your TRNG has a uniform distribution and can generate all possible 32bit numbers (ie. isn't limited to a specific smaller number range, just to [0,2^32-1]).

In this case, it doesn't really matter, because the TRNG output is also a stream of truly random single bits, as well as a stream of truly random 2bit pairs (numbers 0-3), etc.
If this were not true, ie. your 32bit numbers are uniformly random but the single bits are not, it wouldn't be possible: Take the first 32bit number, then see what bits it has, and you already have information about some bits of the next numbers, meaning the full number isn't ranodm either.

So, if you take just a part of one 32bit dataset, it doesn't mean that the rest is less random anymore. Just use the rest too.