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The key to answering this is to understand that Winthrop was a Puritan with a Puritan's view of corruption.
Like most members of the Colony, Winthrop was a Puritan. This group claimed that the Church of England was corrupted by selfish leaders and petty squabbles. In contrast, Puritans envisioned an idealized community in which all citizens would focus their lives on the word of God. Ironically, the Puritans' almost single-minded pursuit of a perfected society based on biblical teachings resulted in impressive success in secular affairs.sjsu.edu
Although one could certainly find examples of Anglican behavior that was less than holy (search any English novel), the key is that the Puritans are zealots - any activity other than the pursuit of purity is corruption. Sports were forbidden as were holidays. (If I had time I'd look up the list of activities forbidden in James' England - I think history.com.uk has a list somewhere, but I find their search function to be as difficult as searching primary sources.)
@jamesqf offers an insightful comment that I think ought to be included:
WRT "the pursuit of purity", the OP might also need to do some research into what Christianity in general, and the Puritan sect, actually meant by purity. I doubt most modern people would understand it.
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Between 1629 and 1640, 20,000 Puritans left England for America to escape religious persecution. They hoped to establish a church free from worldly corruption founded on voluntary agreement among congregants. GilderLehman which is a generally a fantastic source for Early American history.
@Gidds offers this observation that I wanted to include.
I've heard it argued that “to escape religious persecution” would be better rendered more along the lines of “to practise their own religious persecution unhindered”
I've folded in both comments because I want to give credit to the respective authors, and because these highlight the challenge of this question/answer - the words must be understood in the context of Puritan assumptions about the world, and Puritan attitudes towards the world. This is not a simple question about what did a word mean; real understanding requires a close examination of what these words meant to these people at that time; that in turn requires examination of some assumptions that are fairly alien to most modern reader.
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The Protestant Reformation was presented as a reaction to the hypocrisy, worldliness, and corruption of the Catholic Church, but that wasn't good enough for Puritans like Winthrop. He considered that the Church of England, having come into power, had slipped in to the same pattern.
The phrase
where men strain at gnats and swallow camels
suggests that Winthrop's manifesto is constructed along the lines of Jesus's excoriation of the Pharisees during the Sermon on the Mount in Matthew 23. I won't repeat the whole chapter, but here's a glimpse:
23 Woe unto you, scribes and Pharisees, hypocrites! for ye pay tithe of mint and anise and cummin, and have omitted the weightier matters of the law, judgment, mercy, and faith: these ought ye to have done, and not to leave the other undone.
24 Ye blind guides, which strain at a gnat, and swallow a camel.
25 Woe unto you, scribes and Pharisees, hypocrites! for ye make clean the outside of the cup and of the platter, but within they are full of extortion and excess.
Source: Bible Gateway, KJV Matthew 23
To "strain at a gnat" is to make too much of an unimportant matter, conversely to "swallow a camel" is to let some atrocity pass as unimportant. Winthrop elaborates this with
use all severity for maintenance of caps and like accomplishments, but suffer all ruffian-like fashions and disorder in manners to pass uncontrolled
Not that this phrase explains the totality of the "corruption" that Winthrop denounces; it just provides a link. On the other hand, it's probably the key to understanding the whole thing, because it's just a polemical way of saying "my values are better than your values".