How do you pronounce "1–10 + some counter"? I have seen lists of common counters, but they may not contain less common counters, such as ～話 or ～膳.
How do I know how to count with rare counters?
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Comprehensive information can be found in a dictionary such as 数え方の辞典. (The first page is quoted in this answer.)
Basically, there are three types of numbers used with counters
derived from Chinese, native Japanese, or derived from English, respectively. There is no hard rule which to use with a given counter, although as a rule of thumb,
The rule of thumb applies to ～話 and ～膳 that you mention in your question: both a on'yomi, both are counted with Chinese-derived numbers.
One word of caution, though: I would say that even with those counters that are counted with ひと、ふた、… numbers 5, 6, 8, 9, 10 (and higher) are rarely used and Chinese-derived numbers are used instead, i.e. you'd rarely hear 五皿【いつさら】, even though it would be 一皿【ひとさら】.
I'm not sure but the following site could be helpful for you.
You can input Japanese sentence and click the either speaker icon, then you can hear the voice of Japanese. At the top of the site there are language selection, so you have to choose Japanese to hear the Japanese speech.