That's partly correct and partly incorrect. It's incorrect in the claim that the government withheld the potato supply. The potato crop massively failed due to potato blight. The government had nothing to do with the potato shortage. This was not just in Ireland, but also through Europe. It was more significant in Ireland, though, because something like 40% of the population depended on potatoes for most of their nutrition.
It is correct that there were plenty of crops, but these were grain crops, not potatoes, which were raised largely for export. In earlier famines the government had banned exports and made those non-potato crops go to feeding the Irish. That was very unpopular with businesses. In the Great Famine the government did not curb exports, and so even though Ireland was producing enough non-potato food to feed those who had been depending almost solely on potatoes, that food was exported instead. So in that sense, much of the blame for results of the potato crop failures can be laid at the hands of the government.
Edit: there is an interesting song, written by Steven and Peter Jones, based on letters their great-great-grandfather in Ireland sent to their great-grandfather, who had immigrated to the United States. There are five verses, covering the contents of five of the letters, between 1860 and 1892. Although this is after the Great Famine, it provides an interesting look at what life was like in Ireland for ordinary farmers as they went through hardship and had family members leaving the country.
Here's a nice performance of it: https://www.youtube.com/watch?v=QRHQAtKbRTk