My answer goes against the other ones here (including the accepted one).
Before I explain why, however, I commend you for even thinking about this. If you have identified this as a weakness, and something that affects the quality of your lectures, then you are already one step ahead of the game. My experience with most educators in the same situation is that they do not seem to consider it their problem. The mentality seems to be that they've spent years learning the language, and possibly even consider themselves highly proficient in the grammatical sense, having written many academic articles in pristine English, so there's nothing wrong with their English, and if people don't put the effort to understand them then that is their problem. This completely disregards the fact that if as a listener you're expending an unusual amount of focus and effort simply to decipher what is being said, you're unlikely to be able to focus on the nuances of a lecture. My own feeling when I attend lectures where the presenter has a bad pronounciation is that I leave feeling drained and having retained nothing despite having, in theory, deciphered all the words being spoken in the lecture.
Also, I would like to make a distinction between having a strong "accent", and effectively having incorrect "pronunciation". A strong accent that does not harm pronunciation is typically a positive thing, giving a lecturer a unique personality and charm. Therefore, the problem isn't having a regional "accent" per se. An "accent" only becomes a problem when you are actually pronouncing things 'wrongly', forcing your listeners to backtrack and figure out what you meant. Errors in pronunciation (and sometimes even grammar, or unusual expressions), could be due to your own language learning background, making it difficult for others who do not share this background to understand you. The corollary of this of course is that, you will find that people of your own linguistic background will probably find you easier to understand when you mispronounce things the same way they are used to. If your university offers pronunciation training, then it would be something worth looking into.
WHY IT'S A BAD IDEA TO START BY APOLOGISING
I'm not sure about it, because I've read that the first 6 minutes with a new class are the most important ones, and by showing this "weakness" they will conclude that I'm not capable, knowledge-wise, to teach them.
This is 100% spot on, and supported by literature. It's a very bad idea to start your lecture with such an 'apology'. It's absolutely fine (and encouraged) to make students feel safe by encouraging interruptions and asking of questions if something is missed or not understood, but you should separate this from the context of an 'apology' relating to negative first impressions about your own shortcomings!
From personal experience, and as you correctly suspected yourself, I would strongly advise you to avoid any opening statements that directly imprint in the students a lack of quality of what is about to follow. Linking to pedagogical literature, this relates directly to Maslow's hierarchy of needs in terms of feelings of psychological safety, as well as studies showing that teacher attitudes of 'defensiveness' and 'diffidence' can directly affect student motivation and engagement. This is particularly true in the case of externally motivated and extroverted students, who may feel you are actively about to risk their chances of obtaining their external goals, and may speak very vocally indeed about it.
If you start with such an apology, you may be actively sabotaging the rest of the lecture, and possibly even the term, if you fail to recover from that first impression. Furthermore, it may seem to you that such a 'fair warning' is respecting the students, but if you think about it, the students are more likely to feel disrespected, in that they will feel that their personal agenda and goals is being disrespected by being unnecessarily put at risk because of your accent, and that you're basically now telling them they'll have to work and focus twice as hard. I.e. your opening statement will effectively plant in their head the negative thought that "Great, they've lumped me with a teacher who I won't even be able to understand most of the time", starting them off with a negative experience from the outset.
Worse, they may even feel that you're effectively asking for permission to not bother making the effort to speak more clearly than you would have if you hadn't warned them about it!
Your apology can also very easily backfire by turning into a self-fulfilling prophecy, where students will have a lower threshold to losing focus because "they've been warned you'd be hard to understand anyway".
Similarly, it could backfire by instead deterring students from interrupting and asking relevant questions, rather than encouraging them, if they feel uncomfortable that doing so would be effectively perceived by you as being called out on your accent, e.g. "great, the teacher has already shown to be self-conscious about their accent, it was literally the first thing he / she said, I don't want to be the guy that's going to keep pointing their accent out to them by saying I couldn't follow the point they're making".
Instead of words, act. Show you respect your students by actively being aware of it and seeking ways to work around it. E.g.
- while 'reading off slides' is generally not recommended, in your case you could make sure that anything you say that is a crucial point or contains jargon, is always also pointed out in words on a screen
- you could record your lectures and go through the effort of subtitling them after the fact, giving learners the opportunity to revise with subtitles
- you could consider an 'inverted classroom' format which relies more on advanced preparation followed by more personalised inputs at the lab
- ensure you have appropriate formative and summative feedback throughout, so that students can flag their own strengths and shortcomings, and form an impression on the quality of your teaching based on that, rather than any psychological feelings you managed to instill on them during first impressions.
Finally, do continue being aware of it, and ensure you make an effort to speak clearly during your lectures, and if possible, try to seek professional pronunciation coaching in the meantime. The worst thing you could do is apologise at the start, and then do nothing more about it. Your students will lose confidence in you straight away.
PS. Also, I completely disagree with your mentor's advice. If people are having trouble to understand you in the first place, and you counter that by saying you speak more languages than them, all you'll achieve by that is to frustrate them even more. It's totally unnecessary and irrelevant information. Not only will they not 'sympathise', but you risk giving the message that it's not really your weakness for not speaking correctly, but their weakness for not being 'linguistically literate' enough by your standards.