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If the lamp shade is blown out, I am thinking the lamp light is too bright, compared to other available light. If other light is available, to be the main light, the solution might simply be a lower-output bulb in the lamp. Keep in mind that mixed types of light can cause a challenging white balance situation!
I would, myself, probably try to use flash to solve this problem, by letting the flash overpower the lamp. (I use several Canon flashguns, rarely using the camera's pop-up flash.) If, however, the lamp's bulb is the desired main source of light, creative partial blocking of the lamp's bulb, within the shade, between the bulb and camera, while leaving a clear path for the light, from light bulb to subject, might work. Be careful about placing any material too close to a hot light bulb!
This is just an untried guess, on my part: Would it be possible to place a flash unit within or behind the lamp shade, to substitute for the lamp's bulb? This would allow the camera to meter the available light, and the power of the flash could be dialed up or down, until the desired effect is achieved. Ideally, more than one flash unit, if available, would be desirable, to help distribute the light for the most pleasing effect.