One reader left a comment that Taishanese New Yorkers call Manhattan 民铁吾. This is interesting because the transcription only makes sense in Taishanese as the stressed 铁 [hɛt] matches the 'hat' sound well (铁 is  [ti:t] and [tiɛ] in Cantonese and Mandarin respectively). Sounded like [mʌn'hɛ?ŋ̩] in the flow of speech, it is a accurate reproduction of  a native speaker's utterance of Manhattan [mənˈhæʔn̩]. Surely sounds better than the bookish diction 曼哈顿 [ma:n'ha:dɵn].

Symbols t and d, used in various phonetic transcriptions (Cantonese jyutping, Mandarin pinyin, English alphabets, etc), are stand-ins for the IPA voiceless alveolar stops t and tʰ (the h diacritic stands for aspirated). For the rest of this blog I will use t and tʰ as they reveal the underlying phonological relationships more clearly. In addition, Middle Chinese has a 'true' d (voiced alveolar stop).

The Taishanese h-initial is related to the t-initial in Cantonese and Mandarin. An example is 台山-- Toisan in Cantonese but Hoisan inTaishanese. The h is usually accompanied by an overlapping glottal stop1 gesture (ʔh), and the t aspirated (tʰ). There's a related correspondence between zero-initial Taishanese syllables and Cantonese/Mandarin d-initials. The zero-initial is actually a glottal stop ʔ, and d the voiceless stop t. For example, in narrow IPA transcription, 铁/跌 would be ʔhɛt̚ /ʔɛt̚  in Taishanese, tʰi:t̚/ti:t̚  Cantonese, and tʰiɛ/tiɛ Mandarin.

Linguists have reconstructed the alveolar stop shift from Middle Chinese to Cantonese/Mandarin:
d (others tones), t → t
d (ping tone), tʰ → tʰ

This is known as Initial Devoicing (浊音清化) in Chinese historical phonology. The process reduced the number of alveolar stop initials from three (定母d, 端母t ,透母tʰ) to two. Similar change happened to English too. Today, d initials are mostly voiceless (d→t) and t initials aspirated (t→tʰ) in American English although we still use d and t in everyday transcription.

Since Initial Devoicing is thought to have taken place prior to the period of peak migration to the Four Counties, it is reasonable to assume that the t-glottalization is a subsequent development, that is:
Middle Chinese

Cantonese

Taishanese
Examples
d (other tones), t
t
ʔ
跌大打答豆东
d (ping tone), tʰ
ʔh
铁田塔台天吞

Note that preservation of aspiration  in the second step also supports the assumption. We may interpret this as the incomplete production of t, and say that the Taishanese 'relaxed' away the alveolar stops. T-glottalization occurs in English too. For example, the t in button (and Manhattan) is usually replaced by the glottal stop, and 'what' becomes [wɔʔ] sometimes.

Another major source of Taishanese h-initial syllables came from MC h/ɦ (曉母/匣母). Their Cantonese counterpart also has h-initials, i.e. 香(incense), 咸(salty), 好(good), etc.

Taishanese also has t-initial syllables. In general, Taishanese t/tʰ  and Cantonese ts/tsʰ are related. They came from the MC initials dz, ts, and tsʰ (從,精,清母):
Middle Chinese

Cantonese

Taishanese
Examples
dz (other tones), ts
ts
t
谢席习箭进
dz (ping tone), tsʰ
tsʰ
斜墙财千村

Here is a summary  in IPA and 'conventional' transcriptions:

IPA

Conventional-Jyutping-Pinyin


Taishanese
Cantonese

Taishanese
Cantonese
Mandarin

MC
ʔ
t

(Null)
d
d

(),端
ʔh

h
t
t

定(平),透
ʔh
h

h
h
x/h

曉,匣
t
ts

d
z
x/j

從(仄),精
tsʰ

t
c
x/c/q

從(平),清

1A glottal stop [ʔ] is the little pause in the middle of 'uh-oh'.