Hi Janet:
I have never seen that report so I can't say anything about it specifically. I doubt it for the following reasons.
It was impossible for Alexandra to visit Petersburg without being seen by someone. The police did everything possible to prevent visits to Petersburg because it was dangerous. The street traffic was apalling - it was virtually impossible to get through the city because of congestion of carts, wagons and such. The police had nightmares of bombs being tossed from windows along the war and mobs attacking a member of the family in a car stalled on some Petersburg street. The cars broke down a lot so they had to travel with mechanics and extra cars - just in case.
Alexandra would have several choices to get to his flat from Tsarskoe. She could be driven into the city in a private car or an Imperial car. An Imperial car had special plates and could be spotted by anyone. If she went by private car, she still would have needed police protection (she never would have been allowed to travel without it) which means several other cars would have travelled with her. If she went by train from Tsarskoe she could have gone from the public Tsarskoe city station (impossible to do without being observed) or private the private train station. If she left from the priivate train station everyone would have known because their were guards all along the tracks and the train still had to enter the public Tsarskoe Selo train station - and she still would have had to travel by private car from there with police protection - how she could do this without being seen I just don't know. Also, there were police agents at Rasputin's flat and on his street watching everyone who visited him. If Alexandra had gone there to visit him it would have been recorded, unless the spies were called off by the police - and even that would have involved a lot of people to make happen.
For Alexandra to visit Rasputin in Petersburg would have been very hard and not necessary. As you said if she wanted to see him she could at Anna's house in Tsarskoe. Her house is quite near the palace (a few hundred feet) and it would be possible to get there without much hoopla and there would be no need for lots of extra police as they were already there. There were no reporters near by and no traffic on the street that was not allowed by the police. The only people who would know about a visit with Rasputin there were the police, the participants in the meeting and a few servants and court officials.
I also think that Alexandra was very careful about her contact with Rasputin as time went on. Obviously, in the early years he visited the family in at least one palace. A couple of pictures of Rasputin with the girls and Alexandra in their rooms at the Lower Palace at Peterhof have been published. Alexandra wrote Rasputin and they spoke on the phone. Alexandra found Rasputin very hard to understand because he spoke a church-type of Russian to her and he had a heavy Siberian accent. Her Russian was pretty good, but it took her a long time to pick-up this dialect. Rasputin spoke to Alexandra in simple sentences, laced with Slavonic and Bible quotations. Even so, Alexandra had to have Anna there so that she could explain what Rasputin had said to her later in English since she missed a great deal of what he said. These meetings were not easy to pull off and happened infrequently. Most of the time Anna was a runner between the Empress and Rasputin. What wasn't written down or said over the phone could not be copied or listened to by strangers.
One can understand why the Empress ventured out infrequently - it was a big hassle to go anywhere involving dozens - and usually hundreds of people! You couldn't go anywhere without being followed and watched.
So Janet, without having seen the report I can't be sure what I think, but it sounds unlikely to me. Was it published after the revolution? Post February 1917 there were 'interviews' published with Olga and Tatiana where they admitted love affairs with Rasputin and told lurid stories of orgies at the palace. I know that's an extreme case and may not compare to the interviews you mention.
