Can a Moldovan citizen with a UK visa travel to the UK and to Italy?

Yes, if a UK multiple entry Standard Visitor visa has been issued and the time granted (mostly six months) has not been used up. Multiple entries into the Schengen area is also possible based on the 90/180 days rule.

As a general rule multiple entry visas are issued, but exceptions do exist. In cases where a planed visit to the Schengen area is being planned, adding this information in the Additional Information field is advised.

After you arrive in the UK:
Always check for how long leave to enter has been granted

• this may be less than 6 months

After a short stay outside the UK during the leave to enter

• the conditions of the leave of entry may be reconsidered upon reentry

Legal text:

APPENDIX V: Immigration Rules for visitors
...
PART V1. Entry to the UK
...
Types and lengths of visit visa and leave to enter or remain

V 1.5 There are four types of visitor routes which depend on the purpose of the visit: a) Visit (standard) up to 6 months, except...

V 1.6 Within the period for which the visit visa is valid, a visitor may enter and leave the UK multiple times, unless the visit visa is endorsed as a single- or dual-entry visa.

...
PART V9. GROUNDS FOR CANCELLATION OF A VISIT VISA OR LEAVE BEFORE OR ON ARRIVAL AT THE UK BORDER AND CURTAILMENT OF LEAVE Cancellation of a visit visa or leave to enter or remain as a visitor on or before arrival at the UK border

V 9.1 A current visit visa or leave to enter or remain as a visitor may be cancelled whilst the person is outside the UK or on arrival in the UK, if any of paragraphs V 9.2 – V 9.7 apply.

Leave to enter the United Kingdom
...
9.The time limit and any conditions attached will be made known to the person concerned either:
(i) by written notice given to him or endorsed by the Immigration Officer in his passport or travel document; or
...
Suspension of leave to enter or remain in the United Kingdom
10A. Where a person has arrived in the United Kingdom with leave to enter or remain which is in force but which was given to him before his arrival he may be examined by an Immigration Officer under paragraph 2A of Schedule 2 to the Immigration Act 1971. An Immigration Officer examining a person under paragraph 2A may suspend that person’s leave to enter or remain in the United Kingdom until the examination is completed.

Sources:

• Immigration Rules - Guidance
  • Immigration Rules: Index
  • Immigration Rules part 1: leave to enter or stay in the UK
  • Immigration Rules Appendix V: visitor rules
    • as of 2019-10-31
• Standard Visitor visa