Ich überprüfe das Dokument erneut.

überprüfe is the first-person singular (ich) present indicative form of the verb überprüfen. Because überprüfen is a regular (weak) verb, its present-tense paradigm is:

• ich überprüfe
• du überprüfst
• er/sie/es überprüft
• wir überprüfen
• ihr überprüft
• sie/Sie überprüfen

The form überprüfe is simply the stem überprüf- plus the ending -e for "ich."

All three can translate as "to check," but they differ in nuance:

• prüfen: a general "to test" or "to examine" (e.g., eine Maschine prüfen = test a machine).
• überprüfen: "to verify" or "to review carefully," often implying a second look or thoroughness (e.g., den Bericht überprüfen = verify the report).
• kontrollieren: "to inspect" or "to control," often with an official or regulatory sense (e.g., Polizei kontrolliert Fahrzeuge).

erneut means again or once more, often with a slightly formal tone. Compare:

• nochmal / noch einmal (colloquial, "one more time")
• wieder (also "again" but more general for repetition)

Example nuances:

• Ich überprüfe das Dokument erneut. (I’m reviewing the document once more – fairly neutral/formal.)
• Ich schaue es mir nochmal an. (I’ll take another quick look – more casual.)

Yes, German adverbs have some flexibility, though each position has a different emphasis:

• Ich überprüfe erneut das Dokument. (Adverb in the mid-field – slightly more emphasis on erneut.)
• Erneut überprüfe ich das Dokument. (Adverb in the first position – strong emphasis on the repetition.)
• Ich überprüfe das Dokument erneut. (Most neutral: adverb at the end.)

All are grammatically correct; choose the position based on the nuance you want.

There are two separate elements each with its own ü:

1. The prefix über- (meaning “over” or “a­gain” in compounds).
2. The root prüf- from prüfen (“to test/check”), which itself is spelled with an ü.

When combined into überprüfen, both umlauts remain, giving überprüf-, and in the first-person form you write überprüfe.