Not really; C# only supports no-args constructor constraints.
The workaround I use for generic arg constructors is to specify the constructor as a delegate:
public T MyGenericMethod<T>(MyClass c, Func<MyClass, T> ctor) {
// ...
T newTObj = ctor(c);
// ...
}
then when calling:
MyClass c = new MyClass();
MyGenericMethod<OtherClass>(c, co => new OtherClass(co));