Most first-time managers either avoid this ask entirely, or make it the way they'd ask a friend for a favor: informally, underprepared, and hoping goodwill carries it. Budget and headcount requests get evaluated more like a pitch than a conversation, whoever approves it is comparing your ask against every other team's ask, with limited money to go around.
The single biggest mistake is leading with need instead of impact. "My team is stretched thin" is true and also not compelling, everyone's team is stretched thin. What gets funded is a specific, measurable case for what the money or the role produces, and what happens if it doesn't get approved.
"I want to make the case for adding one headcount to the team. Right now [specific gap] is costing us about [X hours/dollars/missed capacity] a month, and it's the direct reason [specific project] slipped last quarter. I've looked at whether we can absorb this with the current team, we can't without cutting into [something else that matters]. If a full hire isn't possible this cycle, I'd also take a contractor for the next two quarters while we build the case for permanent headcount."
"What does this cost us if the answer is no?" is the question you need an honest answer to before you ask, because it's the question they're actually going to ask you.
One more thing worth planning for: know your fallback before you walk in. If the answer is no, or not now, have a follow-up question ready, "what would need to be true for this to get approved next cycle?", so the conversation produces something even when the ask itself doesn't land.