For most spells, no.
As described in the "Areas of Effect" section of the spellcasting rules:
A spell's description specifies its area of effect, which typically has one of five different shapes: cone, cube, cylinder, line, or sphere. Every area of effect has a point of origin, a location from which the spell's energy erupts.
This point of origin is a point in the space from which the magical energy spreads, but it is not its source.
The description of Antimagic Field says, in part (bold for emphasis mine):
Areas of Magic. The area of another spell or magical effect, such as fireball, can't extend into the sphere. If the sphere overlaps an area of magic, the part of the area that is covered by the sphere is suppressed. For example, the flames created by a wall of fire are suppressed within the sphere, creating a gap in the wall if the overlap is large enough.
Hence, if a Fog Cloud is hovering somewhere and an Antimagic Field is cast nearby intersecting the area of effect of the former spell, then the fog is suppressed only in the intersection of the two areas of effect, even if the point of origin is included in this intersection.
For some spells, yes.
There are anyway some spells for which covering the object which is the origin of the spell suppresses the effects on the entire area of effect. For example, consider the Darkness spell that you mentioned (emphasis mine):
If the point you choose is on an object you are holding or one that isn't being worn or carried, the darkness emanates from the object and moves with it. Completely covering the source of the darkness with an opaque object, such as a bowl or a helm, blocks the darkness.
Hence, if Darkness is cast on an object and the Antimagic Field includes that object, the darkness produced by the spell disappears. Indeed, the description clearly states that the origin of the darkness is the object: blocking the magic coming out of the object via an Antimagic Field blocks the effects of the spell.
This suppression is confirmed by another part of the description of the Antimagic Field spell:
Spells. Any active spell or other magical effect on a creature or an object in the sphere is suppressed while the creature or object is in it.
On the other hand, if Darkness is cast choosing a point in space as origin and the Antimagic Field includes such point in its AoE, then the Darkness is supressed (i.e., there is no heavily obscured area caused by the spell) only in the intersection of the two areas.
The same happens for the Light cantrip:
You touch one object that is no larger than 10 feet in any dimension. [...] Completely covering the object with something opaque blocks the light.