(Full image) Beechcraft 1900C.
I cannot find the name of the missing instrument even in the manuals.
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Originally there was no instrument at this place, except for UE series with an electrically driven artificial horizon, where there was a fast erect pushbutton/indicator for the copilot gyro.
According to Wikipedia, UD series is for the military model and UE series is for 1900D, so I assume UC series is for 1900C model. Instrument panel for series UA/UB/UC:
Instrument panel from training manual, models UA/UB/UC (source)
Anything visible here has been added after the aircraft left the plant, when the panel was retrofitted. When browsing pictures of this cockpit online, we can see various additions:
Looking at details of your image, we can see there was likely a sticker and a button associated with the instrument. This could fit with a battery temperature indicator and its test button:
This configuration was actually found on F-GPYY. Comparison:
Full cockpit of F-GPYY when it was an aircraft of the now defunct French Air Littoral operator:
Photo by Thierry Izzi, source
It's a missing instrument because it's not part of the standard avionics of the 1900C model but part of the Beechcraft's 1900D aircraft avionics.
If you take a look at pages 0 and 1 of 31-10-00-01 (chapter 31, section 10, unit 00, figure 01) of the Beechcraft 1900D aircraft's IPC (Illustrated Parts Catalog),¹ you'll notice that the hole featured on the picture that you provided is precisely the item
100 (Part Number
101-380031-3), which is the Copilot's Turn and Slip Electric Indicator (technical nomenclature
INDICATOR-COPILOT TURN AND SLIP, ELECTRIC).
Looks like this 1900C aircraft is using a 1900D aircraft panel plate/harness but it doesn't have all the 1900D's indicators, hence the missing item
¹ Pages 6 and 7 of the PDF file.