DISCLAIMER: I am not responsible for anything you do to your PSP. This will only work if you downgraded with the n00bz or Dark_Alex downgraders or newer. If you don't feel like you could follow this guide or are relatively new at this, please don't attempt it. From what I hear, tem n00bz is working on an automated solution for this so feel free to wait for their release (if the rumor is true of course).

In order to return your TA-082/86 PSP to a state that will allow you to install official Sony Firmware updates, you have to have your idstorage keys the way they were before you used a downgrader. ChillyWilly's keycleaner app will return most keys to their normal state. It usually only leaves 1 key corrupted. The creator of the KeyCleaner app did this on purpose in order to not allow people to go back to official firmware. The key that is need for homebrew is Key 0x0005 and that is the key that ChillyWilly's Keycleaner app leaves corrupted.
Step 1: Clean as many keys as you can using Key Cleaner under OE fw. Restoring keys from the memory stick will ensure all of your keys are restored from the ones that came with your PSP. Restoring them without using the memory stick usually works but it can only restore keys that have been shuffled around, NOT any ones that were deleted. Assuming you were able to restore your keys properly, that leaves just key 5 in the "patched"(ie, the hombrew) state.
Step 2: Revert to 2.71 OFW using recovery mode by placing the 2.71 update in the recovery folder.
Step 3: Next you have to load HEN D in order to enable homebrew to run under 2.71. Now you can run KeyCleaner under HEND and verify that your idstorage key status looks like this:
4 - original key - okay
5 - patched key - okay
6 - original key - okay
8 - original key - okay
8 - no key(code 0x80000025) (this is perfectly normal for newer PSPs)
41 - original key - okay
42 - cleared key - okay
43 - original key - okay
45 - original key - okay
46 - cleared key - okay
47 - original key - okay
Step 4: You can then revert key 5 by running the 1.5 Enabler from the corruption-free 2.71 downgrader package via HEND, then use the hidden "revert" feature - when on the screen that says the PSP has been 1.5 enabled, press SQUARE and it will undo the key 0x0005 patching. You'll be back on the screen saying:
Current Key 5 first longword is 0x436C6B67 1.50 will not run on TA-082/86 PSP with this setting. Press X to alter the key. Press O to exit.
This is exactly what you want.
Step 5: Exit the program and restart your PSP. After that, just update using the network update icon or download the 3.50 update (or any official update for that matter) and you are back on the Sony boat.

EDIT: DON'T FORGET to fix the other idstorage keys using KeyCleaner FIRST!!! This will fix the corruption performed by other downgraders such as the one by Dark_Alex and n00bz.
Here is the package containing HEN D, the 1.5 Enabler and the KeyCleaner
http://www.sendspace.com/file/8fd2sf
Just unzip the contents of the MS_Root folder to your Memory Stick
Here is the 2.71 Recovery folder
http://www.sendspace.com/file/b5abj2
Here is a table of what you keys SHOULD look like in a HEX editor. It is VERY useful if you don't have your original keys anymore.
http://harleyg.net/docs/idtable.txt


Please let me know if you need further clarification or if the links to the files are dead in this thread.

These errors are generated by the hidden updater module sceChkDegeneration, used to detect a patched IdStorage on a TA-082

In order to help you guys out, here's a list of the errors so you can work out what needs to be done to fix your PSP.

DRNFFFFFFD8 = key 0x4 missing
DRNFFFFFFD7 = key 0x4 header is not "n y r B" (in hex: 6E 79 72 42)

DRNFFFFFFCE = key 0x5 missing
DRNFFFFFFCD = key 0x5 header is not "g k l C" (in hex: 67 6B 6C 43)
* to fix this, get KeyCleaner -> HERE <- then let it clean key 5, run it again and tell it to UNPATCH key 5

DRNFFFFFFC4 = key 0x6 missing
DRNFFFFFFC3 = key 0x6 header is not "r d D M" (in hex: 72 64 44 4D)

Additional checks in TA-086:
DRNFFFFFFB9 = key 0x7 header is not "D a P A" (in hex: 44 61 50 41)
DRNFFFFFFB0 = key 0x8 missing
DRNFFFFFFAF = key 0x8 header is not "p D C L" (in hex: 70 44 43 4C)
* to fix this, get IdStorage Manager and


CTA80000025 = region key (0x100) is damaged or missing (irrecoverable unless you have a backup)

Most of these problems can be fixed using the excellent KeyCleaner and IdStorage Manager by Chilly Willy.


Other errors:
FFFFFFFE = critical error at the end of the update - flash1 related, try restoring default settings

Credits to jas0nuk