The internal cleaning procedures usually make the issue worse by compounding it with more ink - you really need to take it apart to get it clean. However, the HP 02 series requires partial disassembly in order to get to the printhead.
One way to do it on the 02 printers is to get a syringe, vinyl tubing and cleaning fluid. You can get a kit on eBay - hook the tube up to the nozzles where the carts are installed. While disassembly works, it’s not something that I’d suggest to end users as there’s too much to get wrong.
The other way is to get empty carts and manufacture cleaning carts (if you can’t buy them), but 02 printers are made of evil because HP enforces the ink expiration date on these and some are bypassable (at your risk) or it’s just not possible so you need aftermarket chips without that BS OR non-expired virgins. In addition to being a nightmare because HP hates you for not buying fresh ink when they want you to based on an “expiration date”*, the lines are usually no better and it needs to be cleaned in both areas anyway - you’re going to need to do it anyway.
*There is some truth to it with inkjets because the ink does eventually dry out or age in a way where it doesn’t print correctly, but it doesn’t happen in the time span HP makes you believe it does. However, the ones where it’s usually mentioned here are disposable and if you kill it, you cry for 5 minutes and buy a new one.