Ten Hag has indeed not made great achievements for Man United but I think he has made quite a significant change. So if he is still trusted to handle Man United for the following seasons, Man United will be a serious challenger in the Premier League. Man United need to get a new striker this season but I don't think they will be able to sign Harry Kane because the price is too high for Man United. Rasmus Hojlund is a pretty good option and the signing could be a great investment for Man United.
Kane wants to play for Bayern and Bayern remain hopeful of getting Kane. Kane's move to Manchester United seems unlikely to me, this is a rival team and even more so I think that he himself will not want to play for Manchester United. If this off-season does not work out with this transfer, then next summer his contract expires and then he can go to Bayern as a free agent, so Tottenham should also be interested in selling him before that time.
Now with Mane going to Al Nassr I think it will make Bayern pursue Kane more.
His current condition will be decisive for Bayern, there are still several players to be used as strikers but with Mane's departure (because he did not enter Tuchel's plans) this will also make them look for other alternatives for strikers and it is not impossible in the next few days Bayern try with a higher price for Kane to be at Bayern for the coming season.
We are now at the end of July, which means there is still one more month to wait about the transfer of players before it is completely closed.