I try to keep the pronunciation as close to the native language's as possible if I am familiar with it.
I try to say "root" for "route" and "foy-ey" for "foyer". Pronouncing "croissant" with the French pronunciation is awkward though, so I usually say it sarcastically. The "e" is "Porsche" always pronounced; even by English pronunciation rules, a vowel preceded by three consosants is pronounced and not silent! Likewise with "Schadenfreude".
I care more about pronunciation and clarity than accent (they are related, but distinct).
I notice things about my own native accent that I don't like, but it sounds a bit weird when I try to change things. For example, I find it ambiguous that I and most Americans use flaps for t's. Latter and ladder, hearty and hardy, leader and liter, let on and led on, metal and medal. By using hard t's in those cases, I end up sounding British or Irish for that one word unintentionally. I also recently learned that New Yorkers distinguish between "cot" and "caught" and between "merry", "marry", and "Mary", which is pretty cool, but I don't think I'm ready to adopt a New York accent.
"hot" may not be exactly the right phoneme (or it's different in your accent), but Brits tend to change the vowel sound between "can" and "can't" and use a hard "t" sound; whereas, American pronounce "can" and "can't" almost exactly the same (with maybe a very short glottal stop at the end for the "t") but just put stress on the negative form (which makes it very hard to tell the difference between the positive and negative sometimes even as a native AE speaker). I personally just try to add a hard "t" at the end to make it clear I'm using the negative form.
I was not planning to enter this conversation, but this post contains many interesting thoughts and points. I grew up in a house with a foyer (with an "r") and never heard any other pronunciation or knew that it was a French word, probably until I went off to college. To me, when a foreign word has been assimilated into (American) English it can, and probably should, take on the (American) English pronunciation.
Root-route, medal-metal, latter-ladder, etc, this is an area where I always differentiate and make sure to clearly say the operative letter.
Can-can't is one place where I place significant emphasis on the "t" to ensure that it is heard. Kind of like the principle in conversation that demands that any "absolute" word such as "always", "never", "only", etc, makes that word the most important word in the sentence.
Merry-Mary-marry is a funny one, I strive to differentiate these but "Mary" ends up slightly closer to "merry" than "marry" when it comes out of my mouth.
Tangential to that topic is the notion of the partial syllable. Perhaps it is something like a sub-dialect but some words contain little glitches that do not rise to the level of being syllables. Like being hinted at but not said. The most common example is the word "often" where I strive to put a tiny hint of a "t" in the middle of it.
There is a city near me, Maryville, that locals say with a very slight central sound that is far less than a syllable but still exists, as mar-(u)-vil, but some people simply cannot hear that central beat (which would be less than a sixteenth note in music). Even after you say it slowly and clearly to them, all they say back to you is mar-vil.
And, of course, there is the capital of Ukraine. Most of us probably grew up with KEY-ev, but when the war started we learned that is the Russian pronunciation. I read numerous articles stating that the word has more than one syllable but less than 2 syllables - so that there is one of those tiny beats before the "v". But somehow many (most?) Americans simply cannot hear the subtlety and say one syllable KEEEV with a hard accent on the long "E" in the middle.
Just in case you are wondering, I grew up in East Tennessee and my mother always told me "There is nothing wrong with being from East Tennessee as long as you don't talk like it."