My iPad continues to say it has been recently updated to iOS 10.3.4

It sounds like you own a fourth generation iPad, which was released by Apple in 2012.

This particular iPad model cannot be updated beyond iOS 10.3.4. Purchase a newer iPad to obtain a more recent iOS version.

Sorry, but this will never be possible as your iPad is too old.

Here's why and what you will need to do about it in 2020!

Tap and read my link below for my additional info about your old iPad model.

If you own OLD 8 and 9 Year Old iPad mo… - Apple Community

Sorry & Best of Luck to You!

If your iPad is currently running, or cannot be updated beyond iOS 10.3.3/4, then have an iPad4.

Update support for iPad4 ended September 2017. The final iOS releases for this model were iOS 10.3.3 (WiFi Only models) and iOS 10.3.4 (WiFi & Cellular). iPad4 cannot be updated to any later major version of iOS as it lacks the minimum hardware requirements needed to run more recent versions.

Many App Developers have also dropped support for older devices and iOS versions preceding iOS11/12/13 - and have withdrawn older versions of their Apps from the Apple App Store. Withdrawal of Apps supporting older models of iPad is making their utility and usefulness significantly more limited. Once compatible versions Apps are withdrawn, you’ll not be able (re)download or (re)install them.

If Apps that you need to use now have higher minimum iOS-version requirements, your only option is to replace your iPad with newer model. Of current models, the entry-level iPad7 supports current iPadOS 13.x (and soon to be released iPadOS 14) and has hardware specifications and capabilities that will vastly outperform your old iPad4. Other models provide a step-up in performance, features and cost.

I hope this information is helpful - if only in clearly stating your current options.