Team.... I am finishing my MSc but it looks like I am gonna have to retake one exam on January to complete it fully. Shall I wait until I finish the MSc completely first and then start a PhD? I think I can prepare for the exam in about two weeks... Thank you

If you can find a supervisor who will let you start the PhD before you have successfully completed your MSc then you should probably start it. However, if you apply now, saying you expect to have to retake one of your exams they may not think that you are that good of a candidate. It may be better to wait until you have completed your MSc and that way potential supervisors need never know that you had a retake.

There are opportunities to start a PhD in January, but, as the team says, your application will look much better once you've completed your MSc

The retake was due to a personal problem rather an accademic failure... I know that a lot of people say this excuse but in my case unforunately it was true... I wish it was just an excuse. Thats why I think it shouldn't matter in my applications for PhDs. Any way i ve got my own funding and the PhD is in entire different subject than my MSc...

As you say, it shouldn't matter in your application, but without the actual qualification you will need very good references from your course tutors stating that you are highly likely to complete the MSc successfully. Even with valid personal reasons for poor performance your potential supervisor needs to know that, had you not had problems to deal with, you would have passed first time. If you believe your references will be good then you might as well begin applying. If your course tutors won't say whether they expect you to pass or fail then you might want to wait until you've got the MSc in the bag.