Time-dependent prize-collecting arc routing problems (TD-PARPs) generalise the regular prize-collecting arc routing problems (PARPs). PARPs have arcs associated with collectable prizes along with travelling costs. TD-PARPs allow travel times to vary at the travelling horizon so that real-life uncertain factors such as traffic and weather conditions can be taken into account. A TD-PARP is to find a travelling route that maximises the profit i.e. total collected prizes minus total travelling costs. TD-PARPs have two facets: selecting a subset of arcs to be travelled and scheduling the selected arcs in the travelling route. TD-PARPs have not been studied much although they are more realistic and generic. In this paper, we first propose a set of deterministic heuristic search algorithms that range from a simple procedure producing quite good results in a fraction of a CPU second to a more extensive procedure producing high-quality results but at the expense of slightly extra CPU time. In this paper, we then propose a meta-heuristic based scatter search (SS) algorithm for TD-PARPs. For the improvement method in the SS algorithm, we propose a multi-operator algorithm that incorporates various neighbourhood operators to diversify the local exploration. For the combination method in the SS algorithm, we propose a 2-level path relinking procedure, which explores combinations of visited and unvisited arcs using two different operators. To control the diversity of the solutions in the SS algorithm, we propose a new effective distance measurement. The experimental results on standard benchmark problems indicate that the proposed SS algorithm significantly outperforms the state-of-the-art existing methods.