It's not a passively cooled cpu.
Without looking at the insides, I would guess there's a big heatsink with a vapor chamber at the bottom or some heatpipes, and a fan on the case that pushes or pulls air through the cooler fins ... so it's not a fan dedicated to cpu only, but a fan that moves air through the case and the cpu cooler fins helping it stay cool.
You have two options :
1. Buy a regular computer case and a regular ATX power supply and an adapter cable which converts the 24pin ATX connector to the connector of your motherboard. Then buy a regular video card, whatever you want.
2. Buy a video card that would work with your existing 240w power supply.
For option 1, you can buy an adapter cable which can make any ATX power supply power your motherboard, here's an example: https://www.ebay.com/itm/PSU-ATX-24pin-to-18-pin-Adapter-Power-supply-cable-HP-Z420-Z620-Z210-Motherboard/123824215473?hash=item1cd47f95b1%3Ag%3AqEEAAOSw9~5ZOT16&LH_BIN=1
So you could buy or use an existing case, buy or reuse an existing ATX power supply and migrate the components to the case. Make sure there's a fan blowing air over the cooler of the cpu, or moving air through the fins of the "passive" cooler as you'd no longer be relying on the case or power supply fan to move air around and keep heatsink cool.
Worst case scenario, you can cut a square hole in the side panel of the case right above where the cpu cooler is and screw a case fan on the side panel blowing air down onto the cpu area.
This way you'd probably be able to buy something with very good performance for the money, like a RX 470 / RX 480 / RX 570 / RX 580 ... the Tech Yes City dude was buying such cards in Australia for 80-100$ a few months ago. There's still good deals on eBay and aliexpress if you don't mind used.
ex. The asian RX 470 4GB for 78$ (a tiny bit slower than regular RX 470) : https://www.aliexpress.com/item/4000180004479.html
or 94$ for RX 480 4GB new - link
or 98$ for RX 570 4 GB - link
For option 2:
Anyway... you have a 240w psu but it's actually the whole 240w available on 12v for components.
The motherboard, cpu, ram and hard drive are probably not consuming more than 100w so you have at least 100-130w to play with.
GT 1030 is really shitty, poor performance for the money. Not worth getting it.
Look for a RX 460 on eBay for around 60$, the versions without additional pci-e 6 pin connector will consume around 50-60w.
Higher performance than GT 1030, almost as good as GT 1050 or better in some games. Here's a couple ebay listings:
2GB Powercolor with bigger fan : https://www.ebay.com/itm/PowerColor-RX460/123879970232
Note that both are full height, if your SFF case only accepts low profile cards it's tough... but if you have room to fit at least the smaller HP model, you could use a pci-e x16 riser cable and simply mount the card somewhere in the case. You could literally cut a cardboard box or a CD/DVD case and make a rectangle big enough for the back of the video card and zip tie the card to that cardboard or plastic to insulate the card from things inside the case, then simply place the video card inside the computer somewhere and use pci-e riser cable to connect card to pci-e slot. As long as you can attach the monitor cable and card has some room for its fan it would work.
There's also RX 550 (~45w) and RX 560 (70-80w) but usually they're quite expensive.
Other alternatives are GT 1050 and Gt 1050ti , which consume around 75w. GTX 1650 also consumes around 80w.
It these cards have a pci-e 6pin connector (some do, just so they won't stress the pci-e slot by taking 60w or more from it ... they split the power taking 30w-ish from slot and 30w-ish from pci-e 6pin connector) it's not the end of the world.
You can buy a 2 x molex -> pci-e 6+2 adapter cable and with a multimeter you can figure out which power supply wires are 12v and which are ground. Then you could simply cut the molex connectors from the adapter cable and connect the yellow wires of the adapter cable (12v) to 12v wires of the power supply, and the black wires (ground) to the ground wires of the power supply (usually black)
You can do it like an amateur and just use a sharp blade to cut a bit of insulation off the psu wires and then wrap new wire to the existing wire and then use electrical tape to insulate everything.
The better than amateur would use a soldering iron to solder the wires together and if you're not afraid of opening the power supply, you could open it and solder the wires directly to the power supply circuit board.
My advice would be to buy a SATA SSD and abandon the mechanical drive ... you'll save around 5w this way and PC will be faster. I'd also suggest buying another stick of 4 GB, or 2 sticks of 2 GB DDR3 (depending on what memory you currently have installed in your PC) You probably have 2 x 2 GB DDR3, for a total of 4 GB in dual channel mode.
Boosting that system to 8 GB of memory will really improve performance. Adding a stick of memory will increase power consumption by about 1-2 watts.
If you're willing to risk it, another option would be even the RX