The spot gold price on your screen usually refers to the price of gold located in London, the most liquid wholesale market for gold traded by financial institutions, electronic exchanges, exchange-traded funds, wholesale jewelry manufacturers, and so on.
When demand falls, the mints often reduce production capacity and add coins to their inventory. As demand rise, these stocks are depleted and production increases. But during these times not all gold coin deliveries are immediate.
Demand for gold coins is considered a retail sentiment indicator for a specific category of investors and as such overall gold coin demand is a miniscule percentage of total gold demand.
In 2020, it was witnessed that whilst gold coin premiums in the US were shockingly high, the gold price in China was trading at a steep discount compared to the London spot price. In other words, if there is a shortage of steel nails in a region, it does not mean that steel prices rise globally.
As gold is a currency, there can't be a shortage of gold for trading. But there can be a shortage of specific gold products at specific locations due to supply chain issues caused by local demand outstripping the supply.
In short, the gold coin price does not have any material bearing on the spot price.