My new fork is considerably smaller where the crown race sits and does not sit in the bike frame snug.
If I cut and install all the way will it work?
My bike is a Carrera Sulcata 1 with a Suntour XCM fork.
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It initially appears as though the crown race from the old fork is still captive in the bike lower headtube. I say this based on the missing crown race in the old fork (though I see marks on the crown of the old fork that may be tool marks from the crown race removal) and in the second photo, within the circle, what one sees occupying the perimeter of headtube looks just like the bottom of a crown race. There would be fitment issues trying to fit a fork with its own crown race into the headtube containing an old crown race. It'd be unusual for a crown race to be stuck (without the fork) in the headtube, but that's what I think I see.
Based on some text in your photos and the photo of the headtube, it seems we are dealing with a Carrera Sulcata 1. The best info I can glean from a few websites that show the specs of this mountain bike is that it sports an 1⅛" semi-integrated headset. These are also known as "Zero-Stack" headsets. Cups that hold the bearings are pushed into the bike's headtube and the bearings reside within aspect of the cup that is pressed into the frame. Most bikes that require "Zero-Stack" headsets have internal diameter headtube measurements (at the ends where the cups and their bearings will reside) of 44mm. The bearings will reduce the openings that the fork steerer tube goes through to 28.6mm or 1⅛ inches and thus, a bike with a complete semi-integrated aka Zero-Stack headset fit forks having a straight 1⅛ inch steerer tube.
All this to say the bike should have the appropriate headset hardware to fit the new forks 1⅛ inch straight steerer tube. The system needs only one crown race, however, so that situation needs to be ironed out.
There is another issue regarding the crown race and lower headset that could impede the proper fit of a new fork (with a new crown race) into an existing headset. The new crown race may have a different taper angle than the headset bearing taper. If there is a mismatch, when sufficient preload is applied (by tightening down the top cap bolt in this system) to remove the play--or the wiggle--of the steer tube in its fit through the headtube and the headset, the steering may bind. It acts like there's too much preload, but any less and the steering has play, so you can't get to a happy medium when there is a mismatch in the angles of the crown race and bearing race.