I'm an international, teaching at an American university, and I have an accent, which can be strong at times. I'm working hard at it, have always been, but still, the accent is there.
This has never been a problem in my career as a researcher because I was most of the time working with senior researchers and grad students. And even if they were native, they've never expressed any concern about my accent.
I'm sure that several times they didn't understand a specific word I was saying but they catch the meaning from the context. Sometimes, people asked me to repeat in an informal way by simply saying "what's that?" or "say it again".
Last year, when I started to teach undergrads, which are unlikely to have experienced different accents, I had some issues. Some students asked me to repeat and I gladly did. My concern is related to those students that didn't ask, either because they are shy or because they thought that by asking I would be offended.
This next semester (yes, I'm going to teach in the summer) I'm thinking to tell my students in the very first minutes of the first class about my accent and tell them that I'm totally fine if they need me to repeat some words.
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.
A mentor of mine told me I should talk openly about my accent, and add that in addition to English, I fluently speak two other languages and can understand two others. Trying then to make a balance between my "weakness" (my accent when speaking English) and my knowledge with languages.