There are several different ways to do this depending on the exact device. However, you need to be careful to match specifications in order to maintain fire safety.
1) if you have a plug-in device, just plug it into a Z wave or zigbee pocket socket.. This is by far the simplest method, and since you didn’t describe the “Spinny light” fully, it wasn’t clear if it’s battery operated, hardwired, or plug-in. But for completeness for others who might find this thread later, we should always mention this option first. Just check the specifications to make sure that the pocket socket can handle the current load. There are many brands and models that will work with SmartThings, the following is just one example. 
https://www.amazon.com/Iris-Smart-Zigbee-Works-Amazon/dp/B01GCKAED4
Or as @rontalley mentioned, if the light already had a Wall switch, you can normally just replace that switch with a smart switch. The exact device will vary depending on the current requirements of the existing switch.
2) for a battery powered device, substitute mains power and then plug it into a pocket socket as in option one
The difficulty of this approach will depend on the type of batteries used by the device. If they are AA batteries, it’s pretty easy to find a dummy battery which you can substitute and create a plug-in option.
https://www.amazon.com/Complete-Adapter-Coupler-Included-Coolpix/dp/B008WYJDPE/
For Devices with other types of batteries, you may have to do some soldering. Know that if you do have to do any soldering, you will certainly void the warranty on the device. But there are still a lot of people who take that path for specific situations.
See The project reports section of the quick browse lists in the community – created wiki and look on the “power” list:
3) if you need to control a mains-powered relay, add a zwave relay, typically a Mimolite or Zooz.
The Mimolite is made specifically to add Z wave controllability to an existing mains powered device. This is a particularly good solution for heavy duty devices. But it may be too expensive for something as simple as a light.
http://www.fortrezz.com/shop/mimo-lite
In the US, The mimolite has become hard to find since the company was acquired by another one which is mostly using the products for professional installs, but in 2019 Zooz introduced a similar device although it’s for lighter loads. It has become quite popular.
4) For Devices with a button or rocker, including existing remotes, use a “robot finger.”. For anything that has a button to be pressed, you can use a tiny actuator. I use these in my own home. And the Switchbot Brand has an official smartthings integration.
This is an easy way to get an integration with an older device, preserves fire safety, won’t void the warranty, and works well. It’s just that it’s expensive, about $40 for the bridge and $29 for each microbot. But it’s a good solution for many retrofit devices were there just aren’t very many other good solutions, and you can now buy them from Amazon.
I wrote a review at the following thread:
2020 Switchbot Review ( robot button pusher), integration through IFTTT or SmartThings app
https://www.amazon.com/SwitchBot-switch-button-controlled-compatible/dp/B07B7NXV4R
https://www.amazon.com/SwitchBot-Hub-Mini-Smart-Remote-dp-B07TTH5TMW/dp/B07TTH5TMW/
5) use an Arduino controller with an ESP8266 Wi-Fi antenna. You have to check specs carefully on these projects to maintain fire safety. The Internet is full of projects where people took five volt devices and hooked them to 3.3 V Arduino’s. Don’t do that. 
But there are various options for dealing with the power requirements safely.
See the following for more discussion of these projects:
6) If the device has its own IR or Bluetooth remote, use a Harmony Hub or other IR Blaster
This is a popular option for the kind of little spotlights that people use on posters in a home theater. These typically have their own handheld remote. If the remote is one that you can replace with a Switchbot mini hub or another smartthings – compatible IR blaster, you can get control that way. At my own house, we use these for some battery operated flameless candles. 
There are some other options if you don’t need to turn the device on and off but are just looking to capture input from it, such as an open/close sensor. But that doesn’t fit your initial description so I won’t detail those here.