Pickering was a Federalist ally of J. Adams, serving in his cabinet after serving in Washington's. Then apparently, had a falling out with Adams (see the Wiki). As a Federalist, he may have had some issues with Jefferson. Long story short, years later, he rekindled his correspondence with Jefferson and used his "unitarianism" as a bridge. He sent Jefferson a copy of William Channing's sermon, that Jefferson apparently had already read.
You can read Jefferson's response to Pickering here where Jefferson proudly identifies as a unitarian, favorably cites Channing along with Joseph Priestley and Richard Price, and of course, slams the Trinity.
Less well known is Pickering's initial letter to Jefferson which you may access here and which I below reproduce in its entirety.
"You will recollect that Gibbon, in his history of the 'Decline and Fall of the Roman Empire,' treats of the Christian Religion, and that he assigns five secondary causes of its prevalence, and final victory over the established religions of the earth. Among these, one was 'the miraculous powers ascribed to the primitive church.' It seems plain that Gibbon considered the miracles, ascribed to Jesus and bis Apostles, alike destitute of reality as those which are found in the legends of the Church of Rome. In relation to the latter, Bishop Watson, in his letters to the historian, puts 'to his heart' this question, 'Whether her absurd pretensions to that very kind of miraculous powers you have here displayed as operating to the increase of Christianity have not converted half her members to Protestantism, and the other half to infidelity?'
"But absurdities in relation to Christianity are not confined within the pale of the Church of Rome. There are some doctrines taught in Protestant churches, in Europe and America, so repugnant to the ideas I entertain of the perfect wisdom, justice, and benevolence of the Deity, as to authorize the opinion that they could not be the subjects of a divine revelation. I have not found them in the books said to contain such a revelation, and I long ago renounced them. They constituted parts of parental and school instruction from my earliest remembrance; but I never taught them to any of my children. I believed them implicitly till I was of age to think and inquire for myself; and one other doctrine to a later period, that of the Trinity, for I had not heard it called in question in any pulpit, and books on the subject had not fallen in my way. Few, indeed, who can read and understand theological controversies allow themselves time to investigate the merits of the questions involved in them. Official and professional duties occupy the attention of most, and, of numbers of the remaining few of educated men, science, and the general pursuits of literature, engross the leisure hours. Some of these to whom doctrines are presented for religious truths which shock their reason, taking them without further inquiry to be the Christian system, they reject this as an imposture."
"I take the liberty, Sir, to send you Mr. Channing's sermon. Whatever you may think of his views of Christianity.
"I am sure that the firm and energetic avowal of his opinions, his candor, his ingenuity, and the elegance of his composition, will fully compensate you for the time you shall spend in its perusal.
"You cannot be uninformed of a prevalent opinion among your fellow-citizens, that you are one of the learned unbelievers in revelation. Your 'Notes on Virginia' contain expressions which, if they did not originate, have served to strengthen, that opinion. You know the influence of a distinguished name over the minds of its warm, and especially of its youthful, admirers; and should you become, if you are not now, a believer, you will deeply regret the effects of that influence. You can entertain no doubt that, eighteen hundred years ago, there appeared in Judea an extraordinary person, called Jesus Christ, the founder of a sect which, after him, were called Christians; for Tacitus, Suetonius, and the younger Pliny speak of him, and of his sect. You also strongly appreciate the moral precepts purporting to have been delivered orally or in writing by Jesus, and by some of his followers who professed to be ear and eye witnesses of his words and the wonderful works ascribed to him. You have called the religion described in the records of those witnesses our 'benign religion;' and could you banish from your mind the recollection of the strange tenets which have been grafted upon that religion, and examine its history and unsophisticated doctrines with the same unbiassed disposition in which you read the histories and other writings of celebrated Romans, you might not think them unworthy to be believed by the most enlightened minds. Certainly, no one can think himself justly exposed to the charge of credulity for entertaining that religious'faith of which Boyle and Locke and Newton were sincere professors.
"A letter from me, unless on business and the common occurrences of life, you would not expect; for to literature I have no pretensions, and in politics we did not agree; but I can disapprove of the principles and oppose the measures of men in public stations with an entire exemption from unkind feelings towards them as individuals. By some I have been injured; but I am not conscious of entertaining a particle of resentment or ill-will towards any human being. In all his imitable perfections, Christians believe it to be their duty to imitate God, ' who' (St. Paul saith) 'will have all men to be Saved, and to come unto the knowledge of the truth.' In this spirit, and in the simple style of antiquity, I bid you, Farewell.
It seems Pickering was concerned Jefferson wasn't a "Christian" and, were he not, was trying to persuade him to convert to pious unitarianism. He basically said, "hey I think the Trinity is an imposture too, but that doesn't stop me from being a devout Christian." Don't throw the baby out out with the bathwater.
But is the Trinity the baby?