While it might seem like a logical opposite of "carefully," English doesn't typically use the prefix "un-" to negate adverbs.
Here are some alternatives to "uncarefully":
* Carelessly: This is the most common and direct replacement.
* Negligently: This implies a lack of care that could lead to harm.
* Thoughtlessly: This emphasizes a lack of consideration.
* Rashly: This suggests acting without thinking things through.
* Recklessly: This indicates a disregard for potential consequences.
The best word to use will depend on the specific context and the intended meaning.