It was no surprise that the dictionary search failed to find any words with this pairing. The /h/ only occurs initially or before a stressed vowel in words such as behind. The /ŋ/ sound only occurs medially or finally, and when medial normally before an unstressed inflection as in singer. Therefore in the standard vocabulary no minimal pairs can occur. In loan words anything is possible, of course. There is a Thai brand of beer called Singha (meaning 'lion'), and this is often pronounced /sɪŋɑ/ by English speakers. (In Thai the final syllable is silent so the beer is called /sɪŋ/ by Thai speakers.) If there were an Indian state of Sihar to rhyme with the state of Bihar, then that might be a minimal pair.
Credit: John Higgins