Does the lymphatic system have a nerve supply and if so, what is its distribution and what is its purpose?

For your information there is a wonderful review article that I call your attention to on this topic (Elenkov IJ, Wilder RL, Chrousos GP, Vizi ES The sympathetic nerve--an integrative interface between two supersystems: the brain and the immune system. Pharmacol Rev. 2000 Dec;52(4):595-638. Please consult that paper for additional details and many helpful references.

My response to your question is primarily derived from this excellent paper.

Innervation of lymph nodes: In lymph nodes, noradrenergic fibers enter at the hilus with the vasculature, and distribute either into a subcapsular nerve plexus or travel with blood vessels through the medullary cords. These fibers run adjacent to both vasculature and lymphatic channels in the medulla and continue with small vessels into the parenchyma of paracortical and cortical regions. Within the nodes a large part of the nerves distribute in the medullary and paracortical areas independently of blood vessels. Noradrenergic fibers supply paracortical and cortical zones (T cell-rich regions) but are absent from nodular regions and germinal centers, the B cell-containing areas. In the human palatine tonsils, noradrenergic fibers distribute along the vasculature to form dense perivascular plexuses and single fibers traveling in parafollicular areas. The epithelium and lymphoid nodules are devoid of noradrenergic fibers.

Innervation of Mucosa-Associated Lymphoid Tissues: Gut-associated lymphoid tissue (GALT) and bronchus-associated lymphoid tissue (BALT) receive both sympathetic and peptidergic innervation. The nerve network within mucosal tissues is very extensive. It has been calculated that the number of nerve cell bodies present in the gastrointestinal tract is equivalent to that found in the spinal cord. Neuropeptides are found in very large amounts in these tissues, particularly SP, vasoactive intestinal peptide (VIP), and somatostatin. Although this issue is not extensively studied, the available information suggests that in GALT including Peyer's patches that represent clusters of lymphoid nodules in the intestines, and the appendix of the rabbit, varicose noradrenergic fibers arborize profusely in the interdomal region of the lamina propria. Here, fibers follow small vessels and branch freely in the parenchyma among fields of lymphoid cells, usually not in association with blood vessels. Nerves predominate in T cell zones of lymphoid aggregates, where they contain neuropeptides and the sympathetic neurotransmitter NE.

Interestingly, as in other lymphoid tissues, intestinal mucosal mast cells lying immediately under the epithelium are apparently selectively associated with enteric nerves, and this is not a random finding. In these tissues, the noradrenergic varicosities are also adjacent to serotonergic enterochromaffin cells. In addition, the nasal mucosa receives tonic discharges from the sympathetic nerves but not from the parasympathetic nerves. Although recent evidence suggests that sympathetic nerve stimulation may up-regulate immunoglobulin (Ig)A secretion in the submandibular glands, as a whole there is almost complete lack of information of how the sympathetic innervation and endogenous CAs may affect mucosal immunity. Interestingly, mucosal immune responses tend to bias toward T helper (Th) 2 responses.