When the shallow network has a small number of neurons, how it will need more sum-product?
Shallowness is when the Model has very few Or just one hidden layer.
When it has a lot of hidden layers it will be a Deep Neural Network.
Sum-product is basically a Neuron of a Hidden Layer.

From the paper-
Artificial neural networks with several hidden layers, called deep neural networks, have become popular due to their unprecedented success in a variety of machine learning tasks (see, e.g., Krizhevsky et al. 2012, Ciresan et al. 2012, Goodfellow et al. 2013, Hinton et al. 2012). In view of this empirical evidence, deep neural networks are becoming increasingly favoured over shallow networks (i.e., with a single layer of hidden units), and are often implemented with more than five layers



A single sum-product hidden layer summarizes a layer of product units followed by a layer of sum units.

The exact definition of Shallow and Deep is a moving number depending on the advancement of AI research.
While it is intuitive that if a 3-layered model needs 5-Neurons in each layer then a single Layered model should have 15 Neurons.
But it will actually require more than that
 It's very difficult to get a conclusive answer to this but the two famous reasons are -

Our data has often an underlying Hierarchical nature i.e. Image data
Analogy with Circuit theory (Check Ref#2 and #4)

Practical reasons seem that it worked with GD/Backprop and research moved in that direction. As you may read the listed references and find that None of the experts has a conclusive view on this.
Reference -
1. Deep Learing Book - 6.4 Architecture Design
2. Neural Networks and Deep Learning,  Michael Nielsen
 3. Stanford CS class CS231n
4. Andrew Ng Deep Learning course - C1W4L04