Jennifer, the ESL online teacher, pronounces ‘seeing a’ as /si: ŋ-gə/ - I expect /si: ŋ-ə/ from dictionaries. So Kindly she brought her captions onto her YouTube channel, I can tell what she was pronouncing. However is the /g/ sound her unique way of saying, or is it one used by many people?
The use of /ŋg/ for 'ng' is a common feature of many accents in the UK. However, it is not something that should be copied by a learner unless they are learning that particular variety of English.
However, I think the reason that this American English speaker said /ŋg/ has nothing to do with her accent. She is speaking unnaturally slowly, and like all people who try to talk to learners in this way, she is using extremely unusual and unnatural intonation patterns and using strong forms where we would expect weak ones.
I strongly suspect that the real reason that she says /ŋg/ here is to do with the sounds on either side of the /ŋ/ combined with the fact she is speaking so slowly. We have a little flap of skin which hangs down at the back of our mouths. If you look into the back of your mouth in the mirror and breathe through your nose, you will see it there. It has a little punch-bag hanging off at the bottom (your 'uvula'). If you look into your mouth in the mirror and say 'aaaaaaaa' you will see it rise up or disappear. This is because, usually, when we speak, this flap called the velum rises to stop air leaving through your nose. It forces the air to leave through your mouth.
With some sounds though - /m, n/ and, of course /ŋ/, - the velum drops down to let air go through your nose. When we have finished this sound the velum goes back up again to block of the route through your nose cavity (usually called your 'nasal cavity').
Because Jennifer is speaking very slowly, and concentrating very hard, the switch from /ŋ/ to the vowel /ə/ [in the word 'a'] isn't happening smoothly. When you make a /ŋ/ sound, your tongue touches the back of your mouth in the same place as it does for /g/. For /g/ your velum is up, and for /ŋ/, your velum is down. Remember that after /ŋ/, the speaker's velum is going to go back up for the next bit of the sentence: 'a chick'. But what happened here is that after her velum went up, her tongue was lagging behind. It was still touching the back of her mouth for the /ŋ/ sound. This turned the /ŋ/ sound into a /g/ for a fraction of a second - before the tongue came back down for the vowel.
This kind of accidentally putting an extra consonant into words is called epenthesis. It is why many speakers produce a /t/ in the word tense so it sounds like tents. Speakers also often insert a /k/ like this after /ŋ/ in the word 'angst', so that we actually hear /æŋkst/.
Hope this is helpful!
I know nothing about American pronunciation of 'ng', but Jennifer's /ŋg/ is not uncommon in Britain. Alan Cruttenden notes in his Gimson's Pronunciation of English (2000):
"Earlier [ŋg] forms are retained instead of RP /ŋ/ in many regional types of speech, notably in the north-west Midlands (e.g. Staffordshire, Derbyshire, Cheshire and south Lancashire), eg singing ['sɪŋgɪŋg] for RP /'sɪŋɪŋg/."