Let's separate the systems into 3 parts: the magnetic material under the coil (1), that not under the coil (2), and the part with/without the gap (3). As current is fed through the coil, (1) magnetizes by forming domains (the electrons start lining up with the field.) Progressively onwards (2) starts magnetizing towards (3). Now, at (3) the gap free section goes about normally. However, the gap version doesn't have a rigid material to magnetize, so the field is weaker.
The fields measured are those standard and induced (pardon my inaccurate terminology.) the standard are from anything but magnetization. Rather than remove the whole chunk, let's just compare two loops of the same ferromagnetic material, but finely carve out half of one to make it half as dense. Throwing a field over them makes the dipoles in both line up. There's only about half as many in one though, so it's induced field will only be about half as strong. The total field is the external plus this.