Is it permissible to carry on eating suhoor when the muezzin is giving the second adhaan, or is that not allowed?
That depends. If the muezzin is giving the adhaan for subh (fajr) (he is giving the adhaan when dawn has really started), then you must stop eating and drinking, because the Prophet (peace and blessings of Allaah be upon him) said: “Do not stop your suhoor when you hear the adhaan of Bilaal, for he gives the adhaan at night, so eat and drink until Ibn Umm Maktoom gives the adhaan.” The basis for this appears in the aayah (interpretation of the meaning):
“and eat and drink until the white thread (light) of dawn appears to you distinct from the black thread (darkness of night)”
If it is known that dawn has come, even without any adhaan, such as when one is in the desert etc., when you see the dawn then you should stop eating and drinking even if you do not hear the adhaan.
But if the muezzin gives the adhaan too early or there is any doubt concerning whether his adhaan has coincided with the dawn or not, then you may eat and drink until you are certain that the dawn has started, whether that is known from a reliable timetable or from a trustworthy adhaan which is known to come at the time of dawn. In this case (if the adhaan is too early) you may eat whilst the adhaan is being given, you may eat or drink what is in your hand, because there is no certainty that the adhaan is being given at the right time, rather it is possible.