What does it mean for the Runner to be “allowed to access 2 or more cards in R&D”? By default, the Runner is allowed to access 1 card in R&D each time they breach R&D.

At the beginning of the breach, the Runner resolves any abilities that grant them additional accesses. If any such ability has already resolved and given the Runner at least 1 additional access, “Pretty” Mary da Silva’s requirement is satisfied, and the Runner can gain another additional access by resolving that ability next.

Can the Runner access an additional card with “Pretty” Mary da Silva if there is an upgrade in the root of R&D?

Not unless they have other additional accesses. Upgrades in the root of a central server are not “in the server” and do not have any effect on “Pretty” Mary da Silva’s requirement.

Mwanza City Grid is rezzed in the root of R&D, and the Runner breaches R&D on their turn with “Pretty” Mary da Silva installed. How many cards does the Runner access in R&D?

The Runner will access 4 cards in R&D (as well as accessing Mwanza City Grid in the root of R&D): the 1 card allowed by default plus the 3 additional cards from Mwanza City Grid’s ability. The Runner cannot gain an additional access from “Pretty” Mary da Silva, because both abilities have the same trigger condition. The Runner resolves their abilities first because it is their turn. Since Mwanza City Grid’s ability is controlled by the Corp, the requirement for the Runner to be allowed to access 2 or more cards is not met until after it is too late to trigger “Pretty” Mary da Silva.