English speakers inserting R in French words

^{This is conjecture, but it's conjecture that's too long to put into the comments.}

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Tl:dr

French /dʀw/ and /tʀw/ are likely to be analysed as /dw/ or /tw/ by English speakers. Because the realisations of /ʀ/ and /w/ in such words are co-articulated and therefore overlap, they mimic what would be analysed as a single /w/ in English. Some English speakers will therefore replicate this pronunciation for all /dw/ and /tw/ sequences.

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Full answer

Doigt and mitoyen

Firstly in a word like droit, there is nothing resembling what an English speaker would recognise as an (English) /r/, the sound [ɹ]. The French /ʀ/, often realised as [ʁ], is unusual enough for English speakers to recognise it in those environments where we would have no comparable sound. However, the [ʁ] in a word like droit is very similar to a sound that many English speakers sometimes encounter when they hear a word-initial /w/ in English, especially if this word is particularly stressed.

English /w/ is a co-articulated consonant. In other words, it involves two strictures (narrowing obstructions of the airflow through the vocal tract). The first is labial, caused by the narrowing and rounding of the lips. The second is velar, cause by the back of the tongue raising towards the velum. In English, it is the labialisation of /w/ which is most prominent, because this stricture is the narrowest. In a language like Spanish, the velar stricture for Spanish /w/ is narrower. It would be fair to say that English speakers find their /w/ more like the vowel /u:/ and Spanish speakers their /w/ more like the velar consonant /g/. However, occasionally, when an English speaker hears an accented or stressed word beginning with /w/ they will hear frication caused by the back of the tongue getting very close to the velum and causing there to be frication, turbulence in the air stream. This is caused by the air from the lungs becoming disturbed as it is forced through the narrow space between the raised back of the tongue and the velum, the soft skin covering the back of the roof of the mouth. In other words the velar aspect of the /w/ will become much more prominent.

Why does this matter? Well, when an English speaker hears a word like droit, they don't hear the beginning of the word as /dʀwɑ/ but just as /dwɑ/. When we speak, our articulators, our mouths, are always preparing for the forthcoming sounds and the articulations necessary to produce them. A French speaker saying droigt will already be rounding and narrowing their lips during the production of the [d] in anticipation of the [w]. In other words the [d] will be labialised. This lip rounding will continue through the /ʀ/, or rather [ʁ], segment. At this point the co-articulation of the French /ʀ/ and the French /w/, will mimic a hyper-articulated English /w/. An English speaker will not recognise the frication as a separate phoneme.

My semi-educated guess (and that's what this is) is that an English speaker's phonological and phonetic brain, when they hear droit, interprets it as /dwɑ/ but detects what they feel to be a specially French articulation of the /w/. They then carry this pronunciation through to all /dwɑ/ sequences they encounter.

One may wonder why an English speaker would interpret a uvular fricative [ʁ] with that component of a hyper-articulated English /w/, the velar fricative [ɣ]. The reason is simply that English has no uvular sounds made at the very back of the mouth. So any fricative made at the back of the mouth counts as the same single type of noise for an English listener (there is no differentiation between velar and uvular sounds).

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Pas

The case with pas is completely different. There are two possibilities that come to mind. The first is that French /ɑ/, [ɑ], is a sound which, in English is associated with words with an orthographic R. For rhotic speakers [those that pronounce English /r/, even when there is no following vowel], it may be that they include an /r/ there because it would be present in English. For non-rhotic speakers [those that only pronounce English /r/ when there is a following vowel], they may be inserting an /ʀ/ in the French word when the following word begins with a non-high vowel. This mimics what non-rhotic English speakers do in English with English /r/.