"a good person"
Your observation is correct. The Finnish letter 'h' corresponds to three different allophones which are explained in Wikipedia like this:
The phoneme /h/ has glottal and fricative allophones. In general, at the end of a syllable it is pronounced as a fricative whose place of articulation is similar to the preceding vowel: velar [x] after a back vowel (/a o u/), palatal [ç] after a high front vowel (/i y/). Between vowels a breathy or murmured /ɦ/ can occur:
-vihma [ˈʋiçmɑ], lyhty [ˈlyçty] -mahti [ˈmɑxti], kohme [ˈkoxme], tuhka [ˈtuxkɑ] -maha [ˈmɑɦɑ]