2 Added alternative example.
source | link

Does this example point you in the right direction?

Example: The following re-links images.

$item = Get-Item -Path "master:\media library\images\koala"
$itemNew = Get-Item -Path "master:\media library\images\penguins"
$links = Get-ItemReferrer -Item $item -ItemLink
foreach($link in $links) {
    $linkedItem = Get-Item -Path master:\ -ID $link.SourceItemID 
    $itemField = $linkedItem.Fields[$link.SourceFieldID]
    $field = [Sitecore.Data.Fields.FieldTypeManager]::GetField($itemField)

    $linkedItem.Editing.BeginEdit()
    $field.Relink($link, $itemNew)
    $linkedItem.Editing.EndEdit() | Out-Null
}

Check out the gist here: https://gist.github.com/michaellwest/f563b0b3597f6c0a75d6

Example: The following gets the list of references to the item.

$item = Get-Item -Path "master:\content\home\sample item 1"
$links = Get-ItemReference -Item $item -ItemLink
foreach($link in $links) {
    # Filter out the links you care about and relink if necessary
}

Does this example point you in the right direction?

Example: The following re-links images.

$item = Get-Item -Path "master:\media library\images\koala"
$itemNew = Get-Item -Path "master:\media library\images\penguins"
$links = Get-ItemReferrer -Item $item -ItemLink
foreach($link in $links) {
    $linkedItem = Get-Item -Path master:\ -ID $link.SourceItemID 
    $itemField = $linkedItem.Fields[$link.SourceFieldID]
    $field = [Sitecore.Data.Fields.FieldTypeManager]::GetField($itemField)

    $linkedItem.Editing.BeginEdit()
    $field.Relink($link, $itemNew)
    $linkedItem.Editing.EndEdit() | Out-Null
}

Check out the gist here: https://gist.github.com/michaellwest/f563b0b3597f6c0a75d6

Does this example point you in the right direction?

Example: The following re-links images.

$item = Get-Item -Path "master:\media library\images\koala"
$itemNew = Get-Item -Path "master:\media library\images\penguins"
$links = Get-ItemReferrer -Item $item -ItemLink
foreach($link in $links) {
    $linkedItem = Get-Item -Path master:\ -ID $link.SourceItemID 
    $itemField = $linkedItem.Fields[$link.SourceFieldID]
    $field = [Sitecore.Data.Fields.FieldTypeManager]::GetField($itemField)

    $linkedItem.Editing.BeginEdit()
    $field.Relink($link, $itemNew)
    $linkedItem.Editing.EndEdit() | Out-Null
}

Check out the gist here: https://gist.github.com/michaellwest/f563b0b3597f6c0a75d6

Example: The following gets the list of references to the item.

$item = Get-Item -Path "master:\content\home\sample item 1"
$links = Get-ItemReference -Item $item -ItemLink
foreach($link in $links) {
    # Filter out the links you care about and relink if necessary
}
1
source | link

Does this example point you in the right direction?

Example: The following re-links images.

$item = Get-Item -Path "master:\media library\images\koala"
$itemNew = Get-Item -Path "master:\media library\images\penguins"
$links = Get-ItemReferrer -Item $item -ItemLink
foreach($link in $links) {
    $linkedItem = Get-Item -Path master:\ -ID $link.SourceItemID 
    $itemField = $linkedItem.Fields[$link.SourceFieldID]
    $field = [Sitecore.Data.Fields.FieldTypeManager]::GetField($itemField)

    $linkedItem.Editing.BeginEdit()
    $field.Relink($link, $itemNew)
    $linkedItem.Editing.EndEdit() | Out-Null
}

Check out the gist here: https://gist.github.com/michaellwest/f563b0b3597f6c0a75d6