I booked a flight from Fiji to the UK via Auckland - Hong Kong - Istanbul - Gatwick. Do I have to claim my baggage in Hong Kong? We will change airlines there from Air NZ to Turkish Air.
You must check your bag tag at the time of initial bag drop for a final answer. The bag tag displays the airport code you will be retrieving your bag.
As @Hilmar answered, if your booking is on different PNRs (Passenger Name Record), ie. separate tickets, the answer is a final yes: you will have to hand your bag for every leg that is covered by a new ticket. Plus, you will pass security check again, as you need to leave for check-in area.
You did not disclose how you bought tickets.
In the case your ticket is a real single ticket bought from a single airline, that depends. If airlines have agreements, and if you don't cross mandatory-customs checkpoints, it's likely that they will check your bag through. This may particularly be the case of subsidiaries (e.g. Turkish Airlines / Anadolu jet, my personal experience).
If you purchased a single ticket by a ticket dealer (travel agency, online dealer like Kiwi.com, see example later) that may not necessarily be the case. Your reseller may be selling you a single ticket that is backed by multiple reservations, thus you fall in the no case. See Kiwi example later. Kiwi is just one of the many online resellers around.
In fact, when I buy extremely cheap flights from certain portals, despite being single tickets, check-through is often not a possibility.
Personal experience: for example, long ago I bought a single RIX->VIE->MXP->ATH
campaign journey for a few euros on Kiwi.com via Skyscanner. I did a single payment and had a single reservation, but while I had explicit guarantee of flight reprotection from Kiwi, I had clear indication both that any checked bag should be retrieved at Vienna and Milan (but I was traveling hand luggage only), and I also had different maximum sizes for hand bag, forcing me to bring a bag of the smallest size allowed (Lauda/Ryanair). This because Kiwi and other online dealers resell cheap tickets combining them into a single reservation.
I just ran another search on Kiwi for the same trip 19th September, random date for convenience of this post. This time, the trip is via Sandefjord, Orio Al Serio. As you can see, on a single ticket they require, and tell in advance, that you have to collect your bag at both airports and self-transfer. Actually, in the case of same airport, self-transfer is a bit odd. A real self-transfer happens when you have to move to another airport, e.g. Heathrow to Gatwitck or Linate to Orio Al Serio, where all transfer is up to you.
The below is explained because Kiwi buys multiple tickets on your behalf
In the end, if your travel choice does not depend on the ability to check the bag through, I would never ask on a travel forum without specific information on the airlines involved, the exact trip plan and the information whether the PNR is single or multiple.
From personal experience of flying Turkish Airlines many times over the past 10 years, always with a stop-over in Istanbul, I never had to get my bags. I would drop them in London - and get them in Tbilisi, or Beijing, or Cairo, etc. Same thing going back.
In fact, once when I missed a connection and had to spend the night in Istanbul (hotel paid for by the airline), I couldn't get my bag even after asking for it.
Note that in my case it has always been a single ticket. If you have different tickets, then you will need to get your bags and re-check them for the second flight.
An exception is when you land in one airport and depart from the other - then it depends on the airline and you should confirm with the airline.