I ran into this issue, found this question, but was not satisfied with the answers because they all seem to suffer if the data you are using contains a delimiter, which should remain quoted.  Getting rid of the unneeded double quotes is a good thing.
The solution below appears to solve this issue for a general case, and for all variants that would cause issues.
I found this answer elsewhere, Removing quotes from CSV created by PowerShell, and have used it to code up an example answer for the SO community.
Attribution: Credit for the regex, goes 100% to Russ Loski.
Code in a Function, Remove-DoubleQuotesFromCsv
function Remove-DoubleQuotesFromCsv
{
    param (
        [Parameter(Mandatory=$true)]
        [string]
        $InputFile,
        [string]
        $OutputFile
    )
    if (-not $OutputFile)
    {
        $OutputFile = $InputFile
    }
    $inputCsv = Import-Csv $InputFile
    $quotedData = $inputCsv | ConvertTo-Csv -NoTypeInformation
    $outputCsv = $quotedData | % {$_ -replace  `
        '\G(?<start>^|,)(("(?<output>[^,"]*?)"(?=,|$))|(?<output>".*?(?<!")("")*?"(?=,|$)))' `
        ,'${start}${output}'}
    $outputCsv | Out-File $OutputFile -Encoding utf8 -Force
}
Test Code
$csvData = @"
id,string,notes,number
1,hello world.,classic,123
2,"a comma, is in here","test data 1",345
3,",a comma, is in here","test data 2",346
4,"a comma, is in here,","test data 3",347
5,"a comma, is in here,","test data 4`r`nwith a newline",347
6,hello world2.,classic,123
"@
$data = $csvData | ConvertFrom-Csv
"`r`n---- data ---"
$data
$quotedData = $data | ConvertTo-Csv -NoTypeInformation
"`r`n---- quotedData ---"
$quotedData
# this regular expression comes from:
# http://www.sqlmovers.com/removing-quotes-from-csv-created-by-powershell/
$fixedData = $quotedData | % {$_ -replace  `
  '\G(?<start>^|,)(("(?<output>[^,"\n]*?)"(?=,|$))|(?<output>".*?(?<!")("")*?"(?=,|$)))' `
  ,'${start}${output}'}
"`r`n---- fixedData ---"
$fixedData
$fixedData | Out-File e:\test.csv -Encoding ascii -Force
"`r`n---- e:\test.csv ---"
Get-Content e:\test.csv
Test Output
---- data ---
id string               notes                       number
-- ------               -----                       ------
1  hello world.         classic                     123   
2  a comma, is in here  test data 1                 345   
3  ,a comma, is in here test data 2                 346   
4  a comma, is in here, test data 3                 347   
5  a comma, is in here, test data 4...              347   
6  hello world2.        classic                     123   
---- quotedData ---
"id","string","notes","number"
"1","hello world.","classic","123"
"2","a comma, is in here","test data 1","345"
"3",",a comma, is in here","test data 2","346"
"4","a comma, is in here,","test data 3","347"
"5","a comma, is in here,","test data 4
with a newline","347"
"6","hello world2.","classic","123"
---- fixedData ---
id,string,notes,number
1,hello world.,classic,123
2,"a comma, is in here",test data 1,345
3,",a comma, is in here",test data 2,346
4,"a comma, is in here,",test data 3,347
5,"a comma, is in here,","test data 4
with a newline","347"
6,hello world2.,classic,123
---- e:\test.csv ---
id,string,notes,number
1,hello world.,classic,123
2,"a comma, is in here",test data 1,345
3,",a comma, is in here",test data 2,346
4,"a comma, is in here,",test data 3,347
5,"a comma, is in here,","test data 4
with a newline","347"
6,hello world2.,classic,123