I would try disconnecting all Hotmail accounts from your Gmail app. Then clearing the cache on your phone which you can instructions for with an email search. Then connect your Hotmail account as an EXCHANGE account which you have to do manually using the server settings that I provided you in an earlier post.
Also, do you have any other email accounts connected to your Gmail app??? If so make sure that you know how to switch back and forth so that you are only displaying the Hotmail account.
And if you have another email account connected to your Gmail app, make sure that you aren't forwarding the Hotmail account emails into the other account.
Since Google added the ability to connect Exchange accounts to the Gmail app for most phones (previously I think it was limited to Nexus phones only) I find that the Gmail app works better than it did before. If I had to, I could use the Gmail account set up as EXCHANGE, but I prefer the Moto Email app which is why I use it.
Gmail with a Hotmail account setup as IMAP -- and the Gmail seems to automatically want to set up a Hotlook account as IMAP unless you manually force it to connect as Exchange -- did not make for a pleasant experience.
You can always try the official Outlook app from Microsoft. What I have personally found is that the official Microsoft app syncs emails without much trouble but it doesn't sync contacts and I hear that the calendar sync is spotty.
If you can't get the Moto Email app in your country (name of the App is "Email") which is what I use and you want to try another app, then I suggest Nine which you can use free for 10 days here in the U.S. and then costs $9.95. I think Nine is the best Microsoft email app, however, I don't use it because I don't have the need for the functionality. But I did try Nine and I liked it.