Update of iPhone SE

Yes, the software upgrade screen reads "A software update is available for your iPhone or iPad. The version of iOS running on your iPhone or iPad does not support wirelessly updating to this version of iOS. Please connect your device to iTunes on a computer to update to the latest software." A link on the screen directs the user to a support page describing using iTunes or the native support on Catalina (which replaced using iTunes). Thank you for the reply.

Well, here is what happened. I have a late 2012 Mac Mini. I replaced the 512GB hard drive with an SSD, which I erased and formatted. I then did the internet recovery to put a new OS on it. I had been running Catalina on the HD but the internet recovery process installed the original OS which was Mavericks onto the SSD. I then tried to download Catalina from the Apple Store to upgrade back. An interesting effect of that is that I had two factor authentication turned on for my AppleId (ie, verification code to trusted cell phone) which cannot be turned off. Mavericks does not know how to accept the verification code. Therefore I had to create a new AppleId that used the old security questions. The net effect is I have a fresh install of Catalina on a new SSD and a new AppleId. BTW, the hard drive vs SSD on the Mac Mini is night and day. I used that new AppleId on the iPhone also. Everything is clean and reset. Now I realize the Mac Mini and the iPhone SE may have support withdrawn any year now. I have a newer iPhone and some iPads that I use for most purposes. The iPhone SE is for a family member. Since Apple will withdraw support at some point I want everything up to date to give me a little extra time after that. The SE is basically the same as a 6S and the 2012 Mac Mini with 16 GB and an SSD runs great for basic tasks like email, web access, and news. It used to take forever to boot and now it is pretty quick.

I came to this forum because I assume I am no longer under my 90 day complimentary support :-). I assume that though everything is "supported" that does not mean Apple does a lot of testing with older versions of software and they would like everything to stay current to minimize testing scenarios. In the case of my Mini I know Apple was not going to add two factor authentication support to Mavericks (even though they they are still downloading it for internet recovery) and I know two factor authentication cannot be turned off but creating a new AppleId with Mavericks that never had it turned on allowed me to get to Catalina and now my Mac Mini is actually pretty nice again. It is like new except with an older 2.5GHz 2 core processor. I'm just thinking iOS 9.3.1 does not have the helper application necessary for the computer update to work as Apple specifies. Otherwise even the iPhone SE is nice, mine is basically new, and some people like the small form factor. I don't know of any other procedure to get software onto the phone or if Apple will do anything about it. I know it is difficult to answer something this obscure but if anyone knows any other means of support or a way of upgrading that will not brick the device I would appreciate it.

You are right, some of that is TMI. I looked at users and groups like you mentioned and on the new install is the new admin I set up.  Under other users I have guest.  If I highlight the new admin I setup I click on password, where I can change the password if I want, and a tab for Login Items.  There is one line item for Login Items and that is iTunesHelper, kind Unknown.  There is a checkbox to hide it, and then a yellow triangle with an exclamation point in it.  The screen has a ? help symbol, and it says that yellow triangle says iTunesHelper won't open because it was moved or deleted.  Thank you for having me look there. I set up a new admin as you suggested and the Login items is empty. I restarted the system so I could use the new second admin I added, plugged in my iPhone, and it works. My phone is updating as I write this. Maybe it has to do with the first admin being set up with Mavericks, and it looks like I should delete and recreate that user since the iTunesHelper has that error symbol. I guess I was assuming the communication error with the helper application was in the iPhone, not in the Mac. Thank you very much for that suggestion.