There is no rule which causes a partially obscured light to cast shadows.
...the DM, of course, can adjudicate otherwise for the sake of realism, because light sources create shadows in real life. There's just no actual rule mandating that people cast shadows.
According to Vision and Light (PHB 183), "shadows" is synonymous with dim light:
Dim light, also called shadows, creates a lightly obscured area.
Now, the light spell in particular creates such an area of dim light at the edge (PHB 255):
Until the spell ends, the object sheds bright light in a 20-foot radius and dim light for an additional 20 feet. The light can be colored as you like. Completely covering the object with something opaque blocks the light.
This description is almost identical to the mundane light source, the candle:
For 1 hour, a candle sheds bright light in a 5-foot radius and dim light for an additional 5 feet.
The lack of any specific rule differentiating magical light from mundane light leaves us with the only possibility that magical light illuminates the same way like normal light (unless otherwise specified; e.g. magical light can be dispelled). Simply, light creates light because a rule says it does, and it doesn't work any differently to nonmagical light because no rule says it does. (Remember also that light only causes an object to glow in way that emits light; it's technically not an area effect spell.)
But does a partially covered or blocked light create additional shadows? Certainly a device exists which allows a partially covered light source to emit light only in one direction: the bullseye lantern (PHB 152), while Darkness and Light (DMG 104-105) implies that opaque physical barriers block light as they would in the real world:
Darkness is the default condition inside and underground complex or in the interior of aboveground ruins, but an inhabited dungeon might have light sources. ... Bright light in an environment of total darkness can be visible for miles, though a clear line of sight over such a distance is rare underground.
But ultimately, there is no actual rule which says a light source casts shadows behind obstacles. Rules-as-written, if you're standing in a 10 foot room with a torch in the middle, the entire room has bright light even if someone is standing between you and the light, or if you're behind a pillar or some other obstacle. The DM can, as always, rule otherwise in the interests of realism, but it's not required.
Best Answer
Destroying an object entirely would destroy the light, because the object no longer exists.
Strictly, the rules do not say what happens when you try to split a light, which means it's up to the DM to adjudicate this situtation.
There's no reason that says part of an object would emit less light. The spell emits the same radius of light whether it's cast on a small object or large one. But breaking the object to get two light sources seems to go against the intent of the rules, which is to create a single light source; you could otherwise illuminate an entire dungeon by lighting up a cake and dropping crumbs as you go.