Given that your SD card reader works well for the normal SD card but gets unrecognized with the microSD cards, it's possible the microSD card slot is damaged or broken. Or, your microSD cards are not properly formatted so they can't work on the iPad. It's suggested to format them on a computer with the right file system.
How to fix the SD card reader not working on iPad
Option 1. Replace the microSD card reader with a new one.
Check if there're any missing components or broken metal contacts on your SD card reader. Since the card reader is not durable good, you can consider buying a new one if the current one keeps failing.
Option 2. Format the microSD card on your PC.
iPad only recognizes camera memory cards formatted as FAT, FAT32, and exFAT. Incompatible file systems can result in your SD card not recognizing on Mac and iOS devices. Here's how to format the SD card on Mac.
Step 1. Connect your microSD card to your Mac via an SD card reader.
Step 2. Open Disk Utility from > Launchpad > Other. Expand on View and choose Show All Devices. On the left column, spot your SD card reader. Under it, click on the microSD card volume.
Step 3. Click on the Erase button on the top toolbar. Assign a disk name and format to your microSD card. You should choose exFAT and MS-DOS (FAT) to make them compatible with your iPad. Keep the scheme unchanged.
Step 4. Click on Erase again to complete the process.
Bonus: How to recover data from formatted SD card
Formatting will give the SD card a new start as well as wipe out all previous data. In case you want to undo the SD card reformat, you can use iBoysoft Data Recovery for Mac to recover deleted and lost data from the SD card.