I have always assumed that Bill and Hillary Clinton's name is pronounced Clin-tun. But during this year's election coverage, I noticed that a great many people pronounce it as Clin-uhn, with no "T" sound at all.
Is there a correct way?
The English language has incredibly many different regional accents, leading to the same words being pronounced differently by different people, sometimes in different places and other times in the same place.
What’s happening here is that some people say [ˈkʰlɪntən], such as Mrs Clinton herself, but others say [ˈkʰlɪ̃ʔn̩] with a glottal stop where the /t/ is phonemically. Those are not the only possibilities, either.
You cannot say that one accent is right and another wrong. They are simply different. It would be unnatural for someone whose accent requires one of those pronunciations to use the other one.
We aren't talking about some people pronouncing a word Fred and other people pronouncing that same word Wilma. Both [ˈkʰlɪntən] and [ˈkʰlɪ̃ʔn̩] use the same underlying abstract phonemes. Those two ways of saying Clinton are just alternate and perfectly valid phonetic realizations of the same phonemes as expressed in two different accents.
Phonemic /t/ has allophones of [tʰ], [t], [ɾ], and [ʔ], depending on a whole lot of factors. Native speakers will not hear a different word if you use a different allophone, because they won’t perceive a phonemic change just because an alternate allophone was used. It’s like did you coming out as didju when spoken rapidly.
Think of these three equivalent scenarios:
Imagine a Scottish speaker named Rory who for /r/ uses a flapped [ɾ] or a rolled [r]. You wouldn’t expect speakers of other accents to reproduce those allophones. It would still be the same word without the fancy /r/.
A name like Norbert or Bernard is always going to be pronounced differently by rhotic speakers as it is by non-rhotic speakers.
A name like Betty is always going to be pronounced differently by speakers who tap that /t/ as a [ɾ] compared with those who put a [t] there.
So both pronunciations of Clinton are perfectly normal ones in their respective dialects. The underlying abstract phonemes are the same; only the allophones are different, which means they are the same words.
Lastly, I would be very leery of expecting folks to reproduce an accent that is not their own. It will easily sound inauthentic, and may well be wrong. You just have to learn not to hear changes to allophones as being changes to the phonemes these represent. This can be especially hard for non-native speakers, because their brains work under different rules.
With names of persons it's generally a good idea to use the same pronunciation as the name bearers themselves. They are the only possible authority regarding the pronunciation of their own name. Especially considering that there is no such thing as a general authority for the English language.
As other comments and the great answer of tchrist have already pointed out Mrs Clinton herself uses [ˈkʰlɪntən]. If you want to consider any pronunciation as correct regarding her name, it's how she herself pronounces it.
Of course, it does not mean that all the other people named Clinton consider that as a correct pronunciation regarding their names or the inhabitants of the cities called Clinton.
I am a (given name) Clinton, my grandfather was also a Clinton. I am from the Great Lakes area of the Midwest (Detroit), though I've lived in the Northeast (Boston area) and in the Southwest US (southern New Mexico). I have family scattered throughout the US.
Clinton is pronounced both ways. I've heard both, and I don't mind either. In the Southwest I tended to hear the "t", in the Northeast the "t" vanishes. In the Midwest, it really is speaker by speaker but mostly it's gone. Native Spanish speakers say the "t" every time.
My own pronunciation? It depends on who I'm talking to and how much I want to blend. But if I'm on the phone with someone who does not know me, I enunciate everything carefully including the "t". (Also to avoid Quentin and Clifton.)
Just naturally? Clin-uhn.
PS: Friends and family call me Clint.
In American English it's pronounced /ˈklɪntən/. The schwa is really, really de-stressed, but the t is definitely there.
I believe speech therapists working with elementary children don't worry about the substitution of a glottal stop for a hard t if it is part of a regional accent. For some children it is a challenge to use the tongue for t in words like kitten and mitten and they may need help if their pronunciation doesn't match family or peers. That said, "Cli-uhn" is jarring to my ears.