"用手推车运" can be translated into English as "transport by handcart" or "carry with a handcart".
Here are some example sentences to illustrate the usage:
We need to transport these materials by handcart to the construction site.
（我们需要用手推车把这些材料运到建筑工地。）
They are carrying the boxes with handcarts to the warehouse.
（他们正用手推车把这些箱子运到仓库。）