The Cecils did not arrange for the Earl of Essex to be sent to Ireland. William Cecil did not use the occasion to replace Essex on the Privy Council with his son.
In the movie Anonymous, William Cecil and his son Robert have a discussion in which William informs his son that he will arrange for the Earl of Essex to be sent to Ireland. The aging William has convinced the Queen to send Essex and to introduce Robert into the Privy Council in the Earl’s place. Soon thereafter, Robert has presumably executed his father’s instructions to arrange for the assassination of Essex and Edward de Vere, the Earl of Oxford, as both assassinations are attempted and fail.
William Cecil dies soon after. The Queen attends the funeral where she is informed by Robert Cecil that Essex has hatched a treasonous plan to gain the crown. She orders that Essex return.
In actual historical fact, the decision to send the Earl of Essex to Ireland was made in early 1599. The politics of the matter are not certain but Essex himself seems to have embraced the opportunity at martial glory. It was his favorite means of gaining status and Ireland the only available theater of action. Had he succeed in bringing Ireland back under English control he would have been the most admired man in England.
William Cecil, towering figure, and closest advisor to the Queen for decades, had already died some six months earlier, in August of 1598. Robert could not have had Essex recalled then as Essex had yet to be appointed Lord Marshall of Ireland much less led his army to the country.
There is no historical record of assassination attempts against either the Earl of Essex or of Oxford. Edward de Vere was, in fact, injured in a street fight, probably around the thigh, but the injury occurred nearly 20 years earlier in an entirely unrelated matter. The Cecils played no role.
As for William Cecil introducing his son to the attention of the Queen, and suggesting that he replace Essex on the Privy Council, Robert had already been a member of the Privy Council since 1591. He was further appointed Principal Secretary to the Queen in 1596, roughly the equivalent of the modern Secretary of State. At that time, William, his father, largely withdrew from active life, unable to perform his duties any longer due to illness and age. William kept his office of Lord Treasurer though the duties were delegated to his son and his treasury staff.