Ask Mr. Smarty Plants

Most often when a white fungus-like substance appears on prickly pear it is an insect infestation not a fungus disease. It is easy to confirm this by seeing if you can scrape off the white fluff and if the white substance produces a bright red juice when it is squeezed between your fingers. If this is the case it is in fact the cochineal insect that is on your prickly pear. The Cactus Doctor, www.thecactusdoctor.com has a discussion about this insect on this website.  He writes, "When you see that your prickly pear and cholla cactus have white sticky mounds covering them, it means you have a cochineal (mealybug) problem. This is not a disease; it is actually an insect infestation. The white sticky mounds are the housing for cochineal bugs (also known as mealybugs). When these white sticky spots first start to appear it is best to spray them off the cactus pads with a power nozzle attached at the end of your hose. If the infestation begins to get out of control, I suggest treating the areas by scrubbing them with insecticidal soap or unscented dish soap. In small areas you can scrub with a toothbrush but for larger areas it is best to use a long-handled brush."

Previously Mr. Smarty Plants answered a question about controlling cochineal insects on cholla cactus (similar to your prickly pears). Here’s what Larry and Brigid Larson wrote:
Feeding cochineals can damage the cacti, sometimes killing their host. 
The Cactus Doctor discusses cochineal eradication.  Their recommendations are:
1)      A power nozzle attached at the end of your hose.
2)      If the infestation begins to get out of control, treating the areas by scrubbing them with insecticidal soap or unscented dish soap was suggested. Neem Oil was also mentioned for a natural approach.

And Nan Hampton wrote the following for another Mr. Smarty Plants question about cochineal bugs on prickly pear cactus. (As you see it is a frequent question.)
Your cactus sounds as if it is infested with cochineal bugs (Dactylopius sp.).  They are small scale insects that feed on the cactus.  They produce fluffy white wax that hides their bodies as they feed on the cactus and protects them from the elements (drying out, in particular) and from predation.  The fluffy wax also serves as a sail or balloon to float on the wind and take the bugs to a new patch of cactus.  The bugs produce carminic acid that also helps protect them from predation, especially from ants. This carminic acid in the bugs has been used by indigenous peoples of southwestern North America, Central America and sub-tropical South America to make a brilliant red dye for centuries, perhaps millennia, to produce beautifully colored textiles. Originally, the cochineal bugs were limited to the New World.  When the European explorers visited and saw the beautiful red cloth of the natives, they took the cochineal bugs back with them and now they occur all over the world.  When a synthetic red dye was produced the demand for cochineal bugs decreased, although they have also been used to help control cactus populations.  Recently, however, after it was determined that the synthetic red dyes can have adverse health side effects, there has been a renewed interest in growing cochineal bugs for red dye.  The dye made from the bugs is currently used in cosmetics and as food coloring.  Because of this, controlling cochineal bugs hasn't really been a priority and, therefore, there isn't a lot of information that I have been able to find for controlling them.  If your infestation is small, I suggest scraping them off (carefully, to avoid the sharp cactus spines) and disposing of them.  You might also be able to wash them off with a water under pressure.  Test a small area first to be sure that you don't injure your cactus and gather up and dispose of any of the insects that you wash off the cactus.

A similar set of solutions are also recommended by the University of Arizona Extension in a publication on Cactus Diseases.

Several webpages mentioned the use of insecticides, and Wikipedia mentioned several natural predators:   “Several natural enemies can reduce the population of the insect on its cacti hosts. Of all the predators, insects seem to be the most important group. Insects and their larvae such as pyralid moths (order Lepidoptera), which destroy the cactus, and predators such as lady bugs (Coleoptera), various Diptera (such as Syrphidae and Chamaemyiidae), lacewings (Neuroptera), and ants (Hymenoptera) have been identified, as well as numerous parasitic wasps."

Here’s some additional information about the fascinating world of the cochineal scale insect called Dactylopius coccus and the carmine dye that was so prized in the 15th century for coloring fabrics. 

Mr. Smarty Plants also offered some further information about cochineal bugs and how they are harvested for natural dyes if this is of interest.