(CNN) Special counsel Robert Mueller's approval rating for handling the Russia investigation stands at 50% in a new CNN Poll conducted by SSRS. That outpaces President Donald Trump's approval rating on the matter by 20 points.

The poll is CNN's first since the conviction of former Trump campaign chairman Paul Manafort on tax and bank fraud charges stemming from Mueller's investigation and guilty pleas from Trump's former lawyer Michael Cohen on several charges after an investigation that Mueller had referred to the US Attorney's Office for the Southern District of New York.

RELATED: Full poll results.

Mueller's approval rating stands at its highest level in CNN polling, rising 3 points since August, a shift within the poll's margin of sampling error. Trump's rating, meanwhile, dipped 4 points, narrowly larger than the error margin.

Cohen's plea addressed charges including those related to his involvement in six-figure payments just before the 2016 election to silence two women for alleged affairs with Trump years previously. The President has denied the claims. The poll finds a growing share of Americans now think Trump did know about the $130,000 payment made directly by Cohen to porn star Stormy Daniels in October 2016 (73% say so in the survey, up from 67% in May). Almost two-thirds in the poll say Trump did direct his lawyer to make that payment, as Cohen claimed in court while entering his guilty plea.

Nearly half of Americans in the poll (47%) say the President ought to be impeached and removed from office, and among those who say Trump did direct Cohen to make the payment to Daniels, 69% say he ought to be impeached. Support for impeachment has risen since June, when 42% said Trump should be removed from office. That increase comes almost entirely among independents -- 48% now say the President ought to be impeached, up from 38% in June -- who have also soured on Trump's job performance generally.

For some, that sentiment may reflect their personal desire for a new president more than an evidence-based assessment of whether he could be impeached. A majority of Americans, 53%, believe there is not enough cause for the House of Representatives to begin hearings into whether Trump should be impeached. Even among those who say the President should be impeached, 20% doubt that there is enough cause to begin hearings in the House.

Democratic leaders have shown no interest, at this time, in pursuing impeachment. House Minority Leader Nancy Pelosi recently said it would be a gift to Republicans, implying it would fire up Trump's base supporters. But Trump has recently warned his supporters that if Democrats take control of Congress next year, it will be a top agenda item for them.

About 6 in 10 Americans say they consider the Russia investigation a serious matter (61%) rather than an effort to discredit Trump's presidency (33%). Around 7 in 10 (72%) say they think Trump should testify under oath for Mueller if asked. Among the 23% who say Trump shouldn't testify, about half feel that way because they think the President has no obligation to do so, while a third think that testifying could be a perjury trap.

The CNN Poll was conducted by SSRS from Sept. 6 through 9 among a random national sample of 1,003 adults reached on landlines or cellphones by a live interviewer. Results for the full sample have a margin of sampling error of plus or minus 3.8 percentage points; it is larger for subgroups.